Square Root Calculator .Co

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Use this online calculator to easily calculate the square root of a given number, including fractions. Quick and easy square root finder.


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When we first encountered the number $i$, we also learned
about complex
numbers, or numbers of the form $a+bi$. 
It
turns
out that square roots of complex numbers are always other complex
numbers.

Consider   $2+3i$  
for the moment.  We can square that number:

Therefore, the square root of   $-5+12i$   is   $2+3i$.  
So now we have demonstrated one case where the square root of a complex
number is another complex number.

Содержание
  1. An algebraic derivation
  2. Epilogue
  3. Square Root 
  4. Prime Factorization Method
  5. Long Division Method
  6. Table for Square and Square root of first ten natural numbers
  7. Square Root Formula
  8. Sample Problems
  9. Simplifying Square Roots
  10. Frequently Asked Questions
  11. Q
  12. How do I find a square root?
  13. A
  14. Q
  15. What are the whole number square roots from 1 to 20?
  16. A
  17. Q
  18. What are numbers with integer square roots?
  19. A
  20. Q
  21. What are the first 20 perfect squares?
  22. A
  23. Q
  24. What are the perfect squares from 1 to 100?
  25. A
  26. Q
  27. How do you determine perfect squares?
  28. A
  29. Q
  30. Why are there no perfect squares between 144 and 169?
  31. A
  32. Practice Questions
  33. What is Square Root?
  34. Repeated Subtraction Method of Square Root
  35. Square Root by Prime Factorization Method
  36. Examples
  37. Square Root by Estimation Method
  38. Examples
  39. Square Root Formula
  40. Square Root of a Negative Number
  41. Examples
  42. Conclusion
  43. Practice Problems
  44. Recommended Reading
  45. FAQs
  46. What is the square root of a number?
  47. How to find the square root of a number?
  48. Can square root be negative?
  49. What is the formula for calculating the square root of a number?
  50. What are the applications of the square root formula?
  51. You May Also Like
  52. CBSE Class 10 Maths Syllabus For 2023-24 (Revised)
  53. CBSE Class 9 Maths Syllabus For 2023-24 (Revised)
  54. CBSE Class 8 Maths Syllabus For 2023-24 (Revised)
  55. Learning Objectives
  56. Solve Quadratic Equations of the form using the Square Root Property
  57. Solve Quadratic Equations of the Form a(x − h)2 = k Using the Square Root Property
  58. Key Concepts
  59. Practice Makes Perfect
  60. Mixed Practice
  61. Writing Exercises
  62. Self Check
  63. Notes:
  64. Square Root:
  65. Square Root Symbol / Square Root Sign:
  66. How are square roots used:
  67. How to Calculate the Square Root of:
  68. Need to calculate the square root of a number with a handheld calculator or looking to buy a new calculator?
  69. Graphing Calculator:
  70. Texas Instruments TI-84 Plus Graphing Calculator
  71. Casio Graphing Calculator (fx-9750GII)
  72. Scientific Calculator:
  73. Texas Instruments TI-36X Pro Scientific Calculator
  74. Math Help / Math Tutor
  75. What is a Square Root?
  76. Properties of Square Root 
  77. Perfect Square 
  78. Methods to find Square Root of a number
  79. 1.  RepeatedSubtractionMethod 
  80. 2.  Prime Factorization Method 
  81. 3.  Division Method 
  82. 4.  Square Roots of Complex Numbers 
  83. Does the calculator support fractions?
  84. Square Root of 1 Solved Examples
  85. How to Find the Square of a Number?
  86. Simplifying Square Root
  87. Square Root
  88. How to Find Square Root?
  89. Repeated Subtraction Method of Square Root
  90. Square Root by Prime Factorization Method
  91. Finding Square Root by Estimation Method
  92. Calculating Square Root by Long Division Method
  93. What is Square Root?
  94. Square Root Definition
  95. Commonly used square roots
  96. What Is the Square Root of 1?
  97. Square Root of 1
  98. Square of a Number
  99. Properties and practical application of square roots
  100. What is a square root?
  101. Square Root of a Negative Number
  102. How calculate a square root
  103. Square Root Formula
  104. Square Root Table
  105. How to Find the Square Root of 1?
  106. Is Square Root of 1 Rational or Irrational?
  107. Squares and Square Roots
  108. Square Root of Numbers
  109. FAQs on the Square Root of 1
  110. What is the Value of the Square Root of 1?
  111. If the Square Root of 1 is 1. Find the Value of the Square Root of 0.01.
  112. What is the Square Root of -1?
  113. FAQs on Square Root
  114. What is Square Root in Math?
  115. How to Calculate the Square Root of a Number?
  116. Can Square Root be Negative?
  117. How to Find the Square Root of a Decimal Number?
  118. What is the Square Root Symbol?
  119. How to Multiply Two Square Root Values Together?
  120. What is the Formula for Calculating the Square Root of a Number?
  121. What is the Square and Square Root of a Number?
  122. Which Method is Used to Find the Square Root of Non-Perfect Square Numbers?
  123. How to Find a Square Root on a Calculator?
  124. What are the Applications of the Square Root Formula?
  125. What does the Square of a Number mean?
  126. How to Calculate the Square Root of a Negative Number?
  127. Why is the Square of a Negative Number Positive?

An algebraic derivation

Epilogue

Algebra is an important topic of mathematics. A square root is an operation that is used in many formulas and different fields of mathematics. This article is about square root and square root formula. The square root of a number is a number squaring which gives the original number. There are multiple square root formulas that are discussed in this article with their problems.

Square Root 

The square root of a number is a number squaring which gives the original number. It is that factor of the number that when squared gives the original number. It is the value of power 1/2 of that number. The square root of a number is represented as √.

Example: Square root of 9 ⇒ √9 = ± 3

                 Here, 32 = 9

                       (-3)2 = 9

The number inside the square root is called radicand and the square root symbol is called radical.

Methods for finding the square root of a number

There are two methods for finding the square roots of a number that are highly used in mathematics. These are discussed below.

  1. Prime Factorization Method
  2. Long Division Method

Prime Factorization Method

Prime factorization is a method in which a number is represented in the form of the product of prime numbers.  Then the square root of the number is fined according to the given concept.

To find the square root using the prime factorization method:

Step 1: Represent the number in its prime factors using prime factorization method.

Step 2: Form the pair of the same factors.

Step 3:Take one factor from each pair and then, find the products of all the factors obtained by taking one factor from each pair.

Step 4:The resultant product is the square root of the number.

Example for Prime Factorization Method:

Long Division Method

Steps to find the square root of a number using the long division method:

Example: Find the square root of 256 using the long division method.

Step 1. Divide the number into pairs starting from one place. For example, pairs starting from one place:= 2, 56

Square Root Calculator .Co

Step 2. After dividing the digits into pairs, start from the leftmost pair or digit. The largest number whose square is just less than or equal to the first pair or digit is taken as the divisor and also the quotient. In the above example, the largest number whose square is just less than 2 is 1. So, the divisor is 1 and the quotient is also 1.

Square Root Calculator .Co

Step 3. Subtract the square of the divisor from the first pair or digit and bring the next pair down to the right of the reminder to get the new dividend. In the above example 2 – 1 = 1 then, we bring the next pair i.e. 56 down and the new dividend becomes 156.

Square Root Calculator .Co

Square Root Calculator .Co

Step 4. Now, the new divisor is obtained by adding the previous divisor and the previous quotient digit and concatenating it with a suitable digit which is also taken as the next digit of the quotient, chosen in a way that the product of the new divisor and this new digit in quotient is equal to or just less than the new dividend. In the above example, the previous divisor is 1 and the previous quotient digit is 1 and their addition gives 2 which is the new divisor. Now, we have to choose a digit so that the product of the new divisor and the new digit in the quotient is equal to or less than the new dividend i.e. 26 is the new divisor and 6 in the new digit which is concatenated with the previous quotient. Now, the current quotient is 16.

Square Root Calculator .Co

Step 5. Repeat steps 2, 3, and 4 till all the pairs have been taken. Now, the resultant quotient is the square root of the given number. In the above example, all pairs have been taken, and hence, the square root of the number 256 is 16. 

Square Root Calculator .Co

Table for Square and Square root of first ten natural numbers

Square Root Formula

The square root of a number has the exponent 1/2. The square root formula is used to find the square root of a number.

Square Root Formula :      √x = x1/2

Sample Problems

Question 1: Find the sum: 5√3 + 6√12

 5√3 + 6√12 = 5√3 + 6(√(4 × 3)

                              = 5√3 + 6 × √4 × √3

                              = 5√3 + 6 ×2√3

                              = 5√3 + 12√3

Question 2: Evaluate:  √64 – √25

 √64 – √25 = √(8 × 8)  – √(5 × 5 )

                          = 8 – 5 = 3

Question 3: Evaluate: √63 / √28

 √63 / √28 = √(7 × 9 )/ √(7 × 4 )

                          = √(9 /4)

                          = √9 / √4 

Question 4: Evaluate: 5 /√15 

 5 /√15 = (5 /√15)×(√15 /√15 ) 

Question 5: Evaluate: 4 / (5 + √6)

                    = (20 – 4√6) / (25 – 6)

                    = (20 – 4√6) / 19

Question 6: Evaluate : 7 / (8 – √10)

                   = (56 + 7√10) / (64 – 10)

                   = (56 + 7√10) / 54

Question 7: Evaluate : (2 +√5) / (4 – √2)

Square Roots and Perfect Squares Video

Hello, in this video, we will explore how to simplify square roots and find perfect squares.

Think you’ve got it? The root symbol is the radical. The index is 4 because it’s the number in the bent arm of the radical. And the radicand is the number under the radical symbol, so in this case, 200.

  4 – index

  200 – radicand

What are the square roots of 36, 81, and 144? Pause the video and try these on your own. When you’re finished, we’ll look over them together.

We get these nice, pretty numbers because we are taking the square roots of perfect squares. But what if we aren’t given a perfect square? Well, then we will have to simplify the square root.

Simplifying Square Roots

When simplifying a square root, we will get all the perfect squares out from under the radical and whatever is remaining from the factors of the radicand stays under the radical. Let’s look at an example.

There are no more perfect squares to take out, so we simply multiply these numbers together to get our new radicand.

Now I want you to try one. Simplify the square root of 96: 96. Pause the video here and simplify. When you’re done, we’ll take a look at it together.

Think you’ve got it? First, we need to find the factors of 96.

I hope this video on square roots and perfect squares was helpful. Thanks for watching, and happy studying!

Frequently Asked Questions

Q

How do I find a square root?

A

Q

What are the whole number square roots from 1 to 20?

A

Not all numbers will have whole number square roots. For example, 1 has a square root because 1 can be expressed as the product of two equal integers, 1ｘ1. 4 has a whole number square root because it can be expressed as the product of 2ｘ2. However, notice how 2 and 3 will not have whole number square roots. 2 cannot be expressed as an integer multiplied by itself. Similarly, 3 cannot be expressed as an integer multiplied by itself.

Q

What are numbers with integer square roots?

A

Numbers with integer square roots are called perfect squares. For example, 64 is a number with an integer square root. 64 is the product of \(8\times8\). When a number can be created by multiplying an integer by itself, it is called a perfect square. Examples of numbers with integer square roots are 1, 4, 9, 16, and 25. All of these numbers have integer square roots. 25 is a perfect square which is the product of \(5\times5\). This means that 5 is the square root of 25.

Not all numbers will have integer square roots. For example, 5 does not have an integer square root. No integer times itself has a product of 5. We can get close: \(1\times=1\), \(2\times2=4\), and \(3\times3=9\), but 5 is not a perfect square, so it does not have an integer square root.

Q

What are the first 20 perfect squares?

A

Q

What are the perfect squares from 1 to 100?

A

There are only ten perfect squares from 1 to 100. 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100. These perfect squares are the result of multiplying a number by itself. For example, 1 is a perfect square because \(1\times1=1\). 4 is a perfect square because \(2\times2=4\). 9 is a perfect square because \(3\times3=9\). Not all numbers are perfect squares. It can be helpful to think of perfect squares as actual squares with a length and a width. For example, 100 is a perfect square built from side lengths of 10 and 10. \(10\times10=100\), therefore 100 is a perfect square.
a 10x10 square

Q

How do you determine perfect squares?

A

Numbers that are considered perfect squares are the result of multiplying an integer by itself. For example, 25 is a perfect square because it is the product of \(5\times5\). A number multiplied by itself creates a perfect square.

Q

Why are there no perfect squares between 144 and 169?

A

We cannot create a perfect square by multiplying something like \(11.5\times11.5\) because it is not the product of two equal integers. The square roots of perfect squares need to be integers.

Practice Questions

 
Which set of numbers contains all perfect squares?

33, 99, 55, 66

33, 99, 55, 66

36, 9, 25, 100

81, 36, 25, 41

The correct answer is 36, 9, 25, 100. Thirty-six is a perfect square composed of \(6×6\), nine is a perfect square composed of \(3×3\), twenty-five is a perfect square composed of \(5×5\), and one hundred is a perfect square composed of \(10×10\).

 
Which pair shows a correct match between the perfect square and its whole number square root?

 
What is the square root of 49?

The correct answer is 7. The square root of 49 is 7, because \(7×7=49\). This also means that 49 is a perfect square.

 
Which value is NOT a perfect square?

The correct answer is 99. Because there is no whole number that can be multiplied by itself to equal 99, it is not a perfect square. Eighty-one is a perfect square composed of \(9×9\), one hundred is a perfect square composed of \(10×10\), and sixty-four is a perfect square composed of \(8×8\).

 
Which pair shows an incorrect match between the perfect square and its whole number square root?

Return to Pre-Algebra Videos

This post is also available in:

Let’s understand what is a square root of a number, how to find the square root of a number, and what are the properties of the square root of a number.

What is Square Root?

The square root of a number is that factor of a number which when multiplied by itself gives the original number. Squares and square roots are special exponents. 

The square root of a number is the inverse operation of squaring a number. The square of a number is the value that is obtained when we multiply the number by itself, while the square root of a number is obtained by finding a number that when squared gives the original number.

If square of a number $a$ is $b$, then square root of the number $b$ is $a$.

It is very easy to find the square root of a number that is a perfect square. Perfect squares are those positive numbers that 

  • Repeated Subtraction Method of Square Root
  • Square Root by Prime Factorization Method
  • Square Root by Estimation Method

Repeated Subtraction Method of Square Root

We subtract the consecutive odd numbers from the number for which we are finding the square root, till we reach 0. The number of times we subtract is the square root of the given number. This method works only for perfect square numbers. Let us find the square root of $36$ using this method.

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$36 – 1 = 35$

$35 – 3 = 32$

$32 – 5 = 27$

$27 – 7 = 20$

$20 – 9 = 11$

$11 – 11 = 0$

Square Root by Prime Factorization Method

Step 1: Divide the given number into its prime factors.

Step 2: Form pairs of similar factors such that both factors in each pair are equal.

Step 3: Take one factor from the pair.

Step 4: Find the product of the factors obtained by taking one factor from each pair.

Step 5: That product is the square root of the given number.

Let’s consider some examples to understand the process.

Examples

Let’s consider some examples to understand the process.

Ex 1: Find the square root of $144$.

Ex 2: Find the square root of $324$.

Square Root by Estimation Method

Estimation and approximation refer to a reasonable guess of the actual value to make calculations easier and more realistic. This method helps in estimating and approximating the square root of a given number. This method is used for numbers that are not perfect squares. 

Let’s consider some examples to understand the process.

Examples

First, find the nearest perfect square numbers to which are less than and greater than $15$. 

These numbers are 

  • $9$ on the lower side since, $3 \times 3 = 9$
  • $16$ on the upper side since, $4 \times 4 = 16$

Let us find the squares of numbers $3.8$ and $3.9$ (numbers closer to $4$). 

Now, repeat the process and check between $3.85$ and $3.9$. 

  • The number of times the process is repeated depends on the number of decimal places required.
  • This is a very long process and time-consuming.

https://youtube.com/watch?v=nEJYRtgHL-4%3Ffeature%3Doembed

Square Root Formula

When $n= 2$, we call it square root. We can use any of the above methods for finding the square root, such as prime factorization, long division, and so on. 

Square Root of a Negative Number

The square root of a negative number cannot be a real number, since a square is either a positive number or zero. But complex numbers have the solutions to the square root of a negative number. 

Let’s consider some examples to understand the process.

Examples

Ex 1: Square root of $-16$.

$-16 = 16 \times \left(-1 \right)$

Conclusion

The square root of a number is that factor of a number which when multiplied by itself gives the original number. There are $3$ methods of finding the square root of a number – Repeated Subtraction Method of Square Root, Square Root by Prime Factorization Method, and Square Root by Estimation Method.

Practice Problems

  1. Find the square root of the following numbers using the repeated subtraction method
    • $25$
    • $121$
    • $196$
  2. Find the square root of the following numbers using the prime factorization method
    • $196$
    • $1296$
    • $784$
  3. Find the square root of the following numbers using the approximation method
    • $31$
    • $53$
    • $90$

FAQs

What is the square root of a number?

How to find the square root of a number?

Can square root be negative?

Yes, the square root of a number can be negative. In fact, all the perfect squares like $4$, $9$, $25$, etc. have two square roots, one is a positive value and one is a negative value. 

For example, the square roots of $4$ are $-2$ and $2$. Since, $\left(-2 \right) \times \left(-2 \right)$ is also equal to $4$. Similarly, the square roots of $9$ are $3$ and $-3$, and so on.

What is the formula for calculating the square root of a number?

What are the applications of the square root formula?

There are various applications of the square root formula:
a) The square root formula is mainly used in algebra and geometry.
b) It helps in finding the roots of a quadratic equation.
c) It is widely used by engineers.

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Quadratic Equations and Functions

Learning Objectives

By the end of this section, you will be able to:

Before you get started, take this readiness quiz.

  1. Simplify: \sqrt{128}.

    If you missed this problem, review (Figure).

  2. Simplify: \sqrt{\frac{32}{5}}.

    If you missed this problem, review (Figure).

  3. Factor: 9{x}^{2}-12x+4.

    If you missed this problem, review (Figure).

A is an equation of the form ax2 + bx + c = 0, where a\ne 0. Quadratic equations differ from linear equations by including a quadratic term with the variable raised to the second power of the form ax2. We use different methods to solve quadratic equations than linear equations, because just adding, subtracting, multiplying, and dividing terms will not isolate the variable.

We have seen that some quadratic equations can be solved by factoring. In this chapter, we will learn three other methods to use in case a quadratic equation cannot be factored.

Solve Quadratic Equations of the form a{x}^{2}=k using the Square Root Property

We have already solved some quadratic equations by factoring. Let’s review how we used factoring to solve the quadratic equation x2 = 9.

\begin{array}{cccc}\begin{array}{}\\ \\ \\ \text{Put the equation in standard form.}\hfill \\ \text{Factor the difference of squares.}\hfill \end{array}\hfill & & & \hfill \begin{array}{ccc}\hfill {x}^{2}& =\hfill & 9\hfill \\ \hfill {x}^{2}-9& =\hfill & 0\hfill \\ \hfill \left(x-3\right)\left(x+3\right)& =\hfill & 0\hfill \end{array}\hfill \\ \begin{array}{c}\text{Use the Zero Product Property.}\hfill \\ \text{Solve each equation.}\hfill \end{array}\hfill & & & \hfill \begin{array}{cccccccc}\hfill x-3& =\hfill & 0\hfill & & & \hfill x-3& =\hfill & 0\hfill \\ \hfill x& =\hfill & 3\hfill & & & \hfill x& =\hfill & -3\hfill \end{array}\hfill \end{array}

We can easily use factoring to find the solutions of similar equations, like x2 = 16 and x2 = 25, because 16 and 25 are perfect squares. In each case, we would get two solutions, x=4,x=-4 and x=5,x=-5.

But what happens when we have an equation like x2 = 7? Since 7 is not a perfect square, we cannot solve the equation by factoring.

Previously we learned that since 169 is the square of 13, we can also say that 13 is a square root of 169. Also, (−13)2 = 169, so −13 is also a square root of 169. Therefore, both 13 and −13 are square roots of 169. So, every positive number has two square roots—one positive and one negative. We earlier defined the square root of a number in this way:

\text{If}\phantom{\rule{0.2em}{0ex}}{n}^{2}=m,\phantom{\rule{0.2em}{0ex}}\text{then}\phantom{\rule{0.2em}{0ex}}n\phantom{\rule{0.2em}{0ex}}\text{is a square root of}\phantom{\rule{0.2em}{0ex}}m.

Since these equations are all of the form x2 = k, the square root definition tells us the solutions are the two square roots of k. This leads to the Square Root Property.

Square Root Property

If x2 = k, then

x=\sqrt{k}\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}x=\text{−}\sqrt{k}\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}x=±\sqrt{k}.

Notice that the Square Root Property gives two solutions to an equation of the form x2 = k, the principal square root of k and its opposite. We could also write the solution as x=±\sqrt{k}. We read this as x equals positive or negative the square root of k.

Now we will solve the equation x2 = 9 again, this time using the Square Root Property.

\begin{array}{cccc}\begin{array}{}\\ \\ \\ \text{Use the Square Root Property.}\hfill \\ \\ \\ \end{array}\hfill & & & \hfill \begin{array}{ccc}\hfill {x}^{2}& =\hfill & 9\hfill \\ \hfill x& =\hfill & ±\sqrt{9}\hfill \\ \hfill x& =\hfill & ±3\hfill \end{array}\hfill \\ & & & \hfill \text{So}\phantom{\rule{0.2em}{0ex}}x=3\phantom{\rule{0.5em}{0ex}}\text{or}\phantom{\rule{0.5em}{0ex}}x=-3.\hfill \end{array}

What happens when the constant is not a perfect square? Let’s use the Square Root Property to solve the equation x2 = 7.

\begin{array}{cccc}& & & \hfill {x}^{2}=7\hfill \\ \text{Use the Square Root Property.}\hfill & & & \hfill x=\sqrt{7},\phantom{\rule{1em}{0ex}}x=\text{−}\sqrt{7}\hfill \end{array}

We cannot simplify \sqrt{7}, so we leave the answer as a radical.

How to solve a Quadratic Equation of the form ax2 = k Using the Square Root Property

x=4\sqrt{3},x=-4\sqrt{3}

y=3\sqrt{3},y=-3\sqrt{3}

The steps to take to use the Square Root Property to solve a quadratic equation are listed here.

Solve a quadratic equation using the square root property.

  1. Isolate the quadratic term and make its coefficient one.
  2. Use Square Root Property.
  3. Simplify the radical.
  4. Check the solutions.

In order to use the Square Root Property, the coefficient of the variable term must equal one. In the next example, we must divide both sides of the equation by the coefficient 3 before using the Square Root Property.

x=7,x=-7

m=4,m=-4

The Square Root Property states ‘If {x}^{2}=k,’ What will happen if <img src="https://pressbooks.bccampus.ca/algebraintermediate/wp-content/ql-cache/quicklatex.com-1abe9c7b6e59710a367f953644a31193_l3.png" alt="k This will be the case in the next example.

c=2\sqrt{3}i,\phantom{\rule{0.2em}{0ex}}\text{}\phantom{\rule{0.2em}{0ex}}c=-2\sqrt{3}i

c=2\sqrt{6}i,\phantom{\rule{0.2em}{0ex}}\text{}\phantom{\rule{0.2em}{0ex}}c=-2\sqrt{6}i

Our method also works when fractions occur in the equation, we solve as any equation with fractions. In the next example, we first isolate the quadratic term, and then make the coefficient equal to one.

x=2\sqrt{10},\phantom{\rule{0.5em}{0ex}}x=-2\sqrt{10}

y=2\sqrt{7},\phantom{\rule{0.5em}{0ex}}y=-2\sqrt{7}

The solutions to some equations may have fractions inside the radicals. When this happens, we must rationalize the denominator.

r=\frac{6\sqrt{5}}{5},\phantom{\rule{0.5em}{0ex}}r=-\frac{6\sqrt{5}}{5}

t=\frac{8\sqrt{3}}{3},\phantom{\rule{0.5em}{0ex}}t=-\frac{8\sqrt{3}}{3}

Solve Quadratic Equations of the Form a(x − h)2 = k Using the Square Root Property

We can use the Square Root Property to solve an equation of the form a(x − h)2 = k as well. Notice that the quadratic term, x, in the original form ax2 = k is replaced with (x − h).

On the left is the equation a times x square equals k. Replacing x in this equation with the expression x minus h changes the equation. It is now a times the square of x minus h equals k.

The first step, like before, is to isolate the term that has the variable squared. In this case, a binomial is being squared. Once the binomial is isolated, by dividing each side by the coefficient of a, then the Square Root Property can be used on (x − h)2.

a=3+3\sqrt{2},\phantom{\rule{0.5em}{0ex}}a=3-3\sqrt{2}

b=-2+2\sqrt{10},\phantom{\rule{0.5em}{0ex}}b=-2-2\sqrt{10}

Remember when we take the square root of a fraction, we can take the square root of the numerator and denominator separately.

\begin{array}{c}\begin{array}{cccccc}& & & \hfill {\left(x-\frac{1}{3}\right)}^{2}& =\hfill & \frac{5}{9}\hfill \\ \\ \text{Use the Square Root Property.}\hfill & & & \hfill x-\frac{1}{3}& =\hfill & ±\sqrt{\frac{5}{9}}\hfill \\ \\ \text{Rewrite the radical as a fraction of square roots.}\hfill & & & \hfill x-\frac{1}{3}& =\hfill & ±\frac{\sqrt{5}}{\sqrt{9}}\hfill \\ \\ \text{Simplify the radical.}\hfill & & & \hfill x-\frac{1}{3}& =\hfill & ±\frac{\sqrt{5}}{3}\hfill \\ \\ \text{Solve for}\phantom{\rule{0.2em}{0ex}}x.\hfill & & & \hfill x& =\hfill & \frac{1}{3}±\frac{\sqrt{5}}{3}\hfill \end{array}\hfill \\ \\ \text{Rewrite to show two solutions.}\phantom{\rule{7em}{0ex}}x=\frac{1}{3}+\frac{\sqrt{5}}{3},\phantom{\rule{0.5em}{0ex}}x=\frac{1}{3}-\frac{\sqrt{5}}{3}\hfill \\ \\ \text{Check:}\hfill & & & & & \\ \text{We leave the check for you.}\hfill & & & & & \end{array}

y=-\frac{3}{4}+\frac{\sqrt{7}}{4},\phantom{\rule{0.5em}{0ex}}y=-\frac{3}{4}-\frac{\sqrt{7}}{4}

We will start the solution to the next example by isolating the binomial term.

\begin{array}{cccc}\begin{array}{}\\ \\ \text{Subtract 3 from both sides to isolate}\hfill \\ \text{the binomial term.}\hfill \\ \text{Divide both sides by 2.}\hfill \\ \text{Use the Square Root Property.}\hfill \\ \text{Simplify the radical.}\hfill \\ \text{Solve for}\phantom{\rule{0.2em}{0ex}}x.\hfill \end{array}\hfill & & & \hfill \begin{array}{ccc}\hfill 2{\left(x-2\right)}^{2}+3& =\hfill & 57\hfill \\ \hfill 2{\left(x-2\right)}^{2}& =\hfill & 54\hfill \\ \hfill {\left(x-2\right)}^{2}& =\hfill & 27\hfill \\ \hfill x-2& =\hfill & ±\sqrt{27}\hfill \\ \hfill x-2& =\hfill & ±3\sqrt{3}\hfill \\ \hfill x& =\hfill & 2±3\sqrt{3}\hfill \end{array}\hfill \\ \text{Rewrite to show two solutions.}\hfill & & & \hfill x=2+3\sqrt{3},\phantom{\rule{0.5em}{0ex}}x=2-3\sqrt{3}\hfill \\ \begin{array}{c}\text{Check:}\hfill \\ \text{We leave the check for you.}\hfill \end{array}\hfill \end{array}

a=5+2\sqrt{5},\phantom{\rule{0.5em}{0ex}}a=5-2\sqrt{5}

b=-3+4\sqrt{2},\phantom{\rule{0.5em}{0ex}}b=-3-4\sqrt{2}

Sometimes the solutions are complex numbers.

\begin{array}{cccc}\begin{array}{}\\ \text{Use the Square Root Property.}\hfill \\ \text{Simplify the radical.}\hfill \\ \text{Add 3 to both sides.}\hfill \\ \text{Divide both sides by 2.}\hfill \\ \\ \text{Rewrite in standard form.}\hfill \\ \\ \text{Simplify.}\hfill \end{array}\hfill & & & \hfill \begin{array}{ccc}\hfill {\left(2x-3\right)}^{2}& =\hfill & -12\hfill \\ \hfill 2x-3& =\hfill & ±\sqrt{-12}\hfill \\ \hfill 2x-3& =\hfill & ±2\sqrt{3}\phantom{\rule{0.2em}{0ex}}i\hfill \\ \hfill 2x& =\hfill & 3±2\sqrt{3}\phantom{\rule{0.2em}{0ex}}i\hfill \\ \hfill x& =\hfill & \frac{3±2\sqrt{3}\phantom{\rule{0.2em}{0ex}}i}{2}\hfill \\ \hfill x& =\hfill & \frac{3}{2}±\frac{2\sqrt{3}\phantom{\rule{0.2em}{0ex}}i}{2}\hfill \\ \hfill x& =\hfill & \frac{3}{2}±\sqrt{3}\phantom{\rule{0.2em}{0ex}}i\hfill \end{array}\hfill \\ \text{Rewrite to show two solutions.}\hfill & & & \hfill x=\frac{3}{2}+\sqrt{3}\phantom{\rule{0.2em}{0ex}}i,\phantom{\rule{0.5em}{0ex}}x=\frac{3}{2}-\sqrt{3}\phantom{\rule{0.2em}{0ex}}i\hfill \\ \begin{array}{c}\text{Check:}\hfill \\ \text{We leave the check for you.}\hfill \end{array}\hfill \end{array}

r=-\frac{4}{3}+\frac{2\sqrt{2}i}{3},\phantom{\rule{0.2em}{0ex}}\text{r}=-\frac{4}{3}-\frac{2\sqrt{2}i}{3}

t=4+\frac{\sqrt{10}i}{2},\phantom{\rule{0.2em}{0ex}}\text{t}=4-\frac{\sqrt{10}i}{2}

The left sides of the equations in the next two examples do not seem to be of the form a(x − h)2. But they are perfect square trinomials, so we will factor to put them in the form we need.

m=\frac{7}{3},\phantom{\rule{0.5em}{0ex}}m=-1

n=-\frac{3}{4},\phantom{\rule{0.5em}{0ex}}n=-\frac{7}{4}

Key Concepts

  • Square Root Property

    How to solve a quadratic equation using the square root property.

    1. Isolate the quadratic term and make its coefficient one.
    2. Use Square Root Property.
    3. Simplify the radical.
    4. Check the solutions.

Practice Makes Perfect

Solve Quadratic Equations of the Form ax2 = k Using the Square Root Property

{a}^{2}=49

a=±7

{b}^{2}=144

{r}^{2}-24=0

r=±2\sqrt{6}

{t}^{2}-75=0

{u}^{2}-300=0

u=±10\sqrt{3}

{v}^{2}-80=0

4{m}^{2}=36

m=±3

3{n}^{2}=48

\frac{4}{3}{x}^{2}=48

x=±6

\frac{5}{3}{y}^{2}=60

{x}^{2}+25=0

x=±5i

{y}^{2}+64=0

{x}^{2}+63=0

x=±3\sqrt{7}i

{y}^{2}+45=0

\frac{4}{3}{x}^{2}+2=110

x=±9

\frac{2}{3}{y}^{2}-8=-2

\frac{2}{5}{a}^{2}+3=11

a=±2\sqrt{5}

\frac{3}{2}{b}^{2}-7=41

7{p}^{2}+10=26

p=±\frac{4\sqrt{7}}{7}

2{q}^{2}+5=30

5{y}^{2}-7=25

y=±\frac{4\sqrt{10}}{5}

3{x}^{2}-8=46

Solve Quadratic Equations of the Form a(x − h)2 = k Using the Square Root Property

{\left(u-6\right)}^{2}=64

u=14,u=-2

{\left(v+10\right)}^{2}=121

{\left(m-6\right)}^{2}=20

m=6±2\sqrt{5}

{\left(n+5\right)}^{2}=32

{\left(r-\frac{1}{2}\right)}^{2}=\frac{3}{4}

r=\frac{1}{2}±\frac{\sqrt{3}}{2}

{\left(x+\frac{1}{5}\right)}^{2}=\frac{7}{25}

{\left(y+\frac{2}{3}\right)}^{2}=\frac{8}{81}

y=-\frac{2}{3}±\frac{2\sqrt{2}}{9}

{\left(t-\frac{5}{6}\right)}^{2}=\frac{11}{25}

{\left(a-7\right)}^{2}+5=55

a=7±5\sqrt{2}

{\left(b-1\right)}^{2}-9=39

4{\left(x+3\right)}^{2}-5=27

x=-3±2\sqrt{2}

5{\left(x+3\right)}^{2}-7=68

{\left(5c+1\right)}^{2}=-27

c=-\frac{1}{5}±\frac{3\sqrt{3}}{5}i

{\left(8d-6\right)}^{2}=-24

{\left(4x-3\right)}^{2}+11=-17

x=\frac{3}{4}±\frac{\sqrt{7}}{2}i

{\left(2y+1\right)}^{2}-5=-23

{m}^{2}-4m+4=8

m=2±2\sqrt{2}

{n}^{2}+8n+16=27

{x}^{2}-6x+9=12

x=3+2\sqrt{3},\phantom{\rule{0.5em}{0ex}}x=3-2\sqrt{3}

{y}^{2}+12y+36=32

25{x}^{2}-30x+9=36

x=-\frac{3}{5},x=\frac{9}{5}

9{y}^{2}+12y+4=9

36{x}^{2}-24x+4=81

x=-\frac{7}{6},x=\frac{11}{6}

64{x}^{2}+144x+81=25

Mixed Practice

2{r}^{2}=32

r=±4

4{t}^{2}=16

{\left(a-4\right)}^{2}=28

a=4±2\sqrt{7}

{\left(b+7\right)}^{2}=8

9{w}^{2}-24w+16=1

w=1,w=\frac{5}{3}

4{z}^{2}+4z+1=49

{a}^{2}-18=0

a=±3\sqrt{2}

{b}^{2}-108=0

{\left(p-\frac{1}{3}\right)}^{2}=\frac{7}{9}

p=\frac{1}{3}±\frac{\sqrt{7}}{3}

{\left(q-\frac{3}{5}\right)}^{2}=\frac{3}{4}

{m}^{2}+12=0

m=±2\sqrt{2i}

{n}^{2}+48=0.

{u}^{2}-14u+49=72

u=7±6\sqrt{2}

{v}^{2}+18v+81=50

{\left(m-4\right)}^{2}+3=15

m=4±2\sqrt{3}

{\left(n-7\right)}^{2}-8=64

{\left(x+5\right)}^{2}=4

x=-3,x=-7

{\left(y-4\right)}^{2}=64

6{c}^{2}+4=29

c=±\frac{5\sqrt{6}}{6}

2{d}^{2}-4=77

{\left(x-6\right)}^{2}+7=3

x=6±2i

{\left(y-4\right)}^{2}+10=9

Writing Exercises

In your own words, explain the Square Root Property.

Answers will vary.

In your own words, explain how to use the Square Root Property to solve the quadratic equation {\left(x+2\right)}^{2}=16.

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table provides a checklist to evaluate mastery of the objectives of this section. Choose how would you respond to the statement “I can solve quadratic equations of the form a times x squared equals k using the Square Root Property.” “Confidently,” “with some help,” or “No, I don’t get it.” Choose how would you respond to the statement “I can solve quadratic equations of the form a times the square of x minus h equals k using the Square Root Property.” “Confidently,” “with some help,” or “No, I don’t get it.”

Choose how would you respond to the statement “I can solve quadratic equations of the form a times the square of x minus h equals k using the Square Root Property.” “Confidently,” “with some help,” or “No, I don’t get it.”

If most of your checks were:

(fill in either box above to explore more about square roots)
Answers are rounded to seven decimal places.

  • positive numbers: 2
  • negative numbers: -3
  • imaginary numbers: -7i
  • complex numbers: 8+9i

Notes:

  • The square root calculator provides the principal square root (the positive square root which is most commonly used).  
    Multiplying the principal square root by -1 will provide the negative square root if needed.
  • The square root of a negative number results in an imaginary number noted by the letter «i».

Square Root:

The square root of a number is a value that when multiplied by itself equals the original number.  
Example:   The square root of 9 = 3.   Thus: 3 x 3 = 9.

Where A and B are valid numbers:
A is the square root of B if:   A x A = B

  • Square Root Symbol / Square Root Sign:

    The square root of a number N is commonly represented by:  

    It can also be represented by:  

    In exponential notation, it can be represented by:   N1/2

    «Square root» can be abbreviated as «sqrt»   Example:   sqrt(2) = 1.4142136

    See: square root symbol if you need to add the symbol to a text document.

  • How are square roots used:

    Square roots can be used:

    • to solve for the distance between two points (Pythagorean Theorem)
    • to solve for the length of a side of a right triangle (Pythagorean Theorem)
    • to find the solutions to quadratic equations
    • to find normal distribution
    • to find standard deviation
    • basically to solve for a squared variable in an equation

    If you see an equation like N2 = 27 , you can solve for N by taking the square root of 27 which
    gives N = 5.1961524

How to Calculate the Square Root of:

  1. A Positive Number:

        The square root of a positive number can be calculated by:

      <!—
    1. SquareRootCalculator.Co website

      2. —>

    2. square root calculator above
    3. handheld calculator
    4. spreadsheet
    5. square roots without a calculator


  2. A Negative Number:

    To calculate the square root of a negative number, find the square root of the same positive number and multiply by «i». ( where i represents an imaginary number and i = square root of -1)

    Example: square root of -5

    = (square root of 5) x (square root of -1)

    = (square root of 5) x (i)

    = 2.236068 x i

    = 2.236068i

  3. An Imaginary Number:

    To calculate the square root of an imaginary number, find the square root of the number as if it were a real number (without the i) and then multiply by the square root of i (where the square root of i = 0.7071068 + 0.7071068i)

    Example: square root of 5i

    = (square root of 5) x (square root of i)

    = (2.236068) x (0.7071068 + 0.7071068i)

    = 1.5811388 + 1.5811388i


  4. A Complex Number:

    To find the square root of a complex number, use the following equation:

    Square root (a + bi) = c + di

    Where:
    c = (1/square root of 2) x square root of [ (square root of(a2+b2)) + a ]

    d = (sign of b/square root of 2) x square root of [ (square root of(a2+b2)) — a ]

    Example: the square root of 3-5i = c + di

    c = (1/square root of 2) x square root of [ (square root of(32+(-5)2)) + 3 ]

    c = 2.1013034
    d = (-1/square root of 2) x square root of [ (square root of(32+(-5)2)) — 3 ]
    d = -1.1897378
    Thus: square root of 3-5i = 2.1013034 — 1.1897378i

Need to calculate the square root of a number with a handheld calculator or looking to buy a new calculator?

Most handheld calculators have a square root function.
Below is a review of some that are recommended and how to use them.

Graphing Calculator:

  • Texas Instruments TI-84 Plus Graphing Calculator

    Square root on TI-84 Plus Calculator

    <!—
    test
    —>

    To take the square root of a number, press [2ND] (the secondary function key) and then [] (the radical symbol key which is used to take the square root of a number) and then the number that you want to find the square root of and then the [ENTER] key.

    (Note: This same method will also work with TI-83 and TI-81 calculators)

    (To see what the graph looks like on this calculator,
    click the show graph button below the calculator image on this page.)

    Pros:
    It can be used on many college entrance exams (check your exam criteria).

    It is a popular calculator. (If you need help, the likelihood of finding someone who knows how to use it is higher).

    The display shows seven lines of input/output. Long equations can be viewed and checked. (This is a nice benefit that graphing calculators have compared to scientific calculators that may only have a single line display.) Another benefit of having a large display is you can compare your current answer to past answers that are still on the screen. This can often help you discover an input error that may have otherwise gone unnoticed.

    Cons:
    It is bulkier than a scientific calculator.
    It costs around $85 (USD) more than a scientific calculator.

    Price:
    Best price for this calculator as of 9-2-2014 is around $94.00 (USD).

  • Casio Graphing Calculator (fx-9750GII)

    Square root on Casio fx-9750GII Calculator

    <!—

    Click to show y =
    √x
     graph

    —>

    <!— test —>

    To take the square root of a number, press [SHIFT] and then [] (the radical symbol is above the x2 key) and then the number that you want to find the square root of and then the [EXE] key.

    (To see this graph, click the show graph button below the calculator image on this page.)

    Pros:
    The cost is half the price of a TI-84 calculator.

    It is a little smaller than the TI-84 calculator.

    It can be used on many college entrance exams (check your exam criteria).

    The display shows seven lines of input/output.

    Cons:

    It is not as popular as the TI-84 calculator. (It may be harder to find someone to help if you have a question about how to use the calculator.)

    Price:
    Best price as of 9-2-2014 is around $42.74 (USD).

Scientific Calculator:

  • Texas Instruments TI-36X Pro Scientific Calculator

    Square root on TI-36X-Pro Calculator

    To take the square root of a number, press [2ND] (the secondary function key) and then [] (the radical symbol key which is used to take the square root of a number) and then the number that you want to find the square root of and then the [ENTER] key.
    Note:  If your calculator is in «Math Print» mode, the display will show as the answer. To convert this answer to a decimal number, press the [<>~] key (the «Answer Toggle» key) which is the key above the [ENTER] key.

    Pros:
    It is good if a graphing calculator is not permitted.
    It is smaller and lighter than a graphing calculator.
    The display shows four lines of input/output. Some scientific calculators only show one line of input/output.

    Cons:
    The symbols shown on the function keys ( + , — , etc.) may be hard to see because they are the same silver color as the keys.

    Price:
    Best price for this calculator on 9-2-2014 is around $19.99 (USD).

Math Help / Math Tutor

If you need help with math:

  • There are a lot of online resources that are free to use and are available when you need them. Some sites have online courses, instructional videos, and even math games to help you learn more about math.
  • Ask someone who is good at math to help. If you can’t find anyone who can help, a math teacher at a local school may be able to help or recommend a math tutor in your area.
  • A local library may be able to help not only with informational books but they may be able to recommend a math tutor also.
  • There are several websites that can help you find a local tutor if you would like in-person tutoring. Also, some tutors are able to help from a remote location if needed. Prices are typically between $10 to $50 (USD) per hour. Always research and try to find reviews, ratings, or feedback from others who have used a particular tutor before making a selection.

In everyday situations, the challenge of calculating the square root of a number is faced. What if one doesn’t have access to a calculator or any other gadget? It can be done with old-fashioned paper and pencil in a long division style. Yes, there are a variety of ways to do so. Let’s start with discussing Square root and its properties first.

What is a Square Root?

A square root is a value, which gives the original number that multiplication of itself. e.g., 6 multiplied by itself gives 36 (i.e. 6 × 6 = 36), therefore, 6 is the square root of 36 or in other words, 36 is the square number of 6.

Suppose, a is the square root of b, then it is represented as,

a = √b or  

a2 = b

Let the square of 2 is 4 so the square root of 4 will be 2 i.e. 

√4 = 2

Hence, the square root of the square of a positive number gives the original number. However, the square root of a negative number represents a complex number.

Properties of Square Root 

  • A perfect square root always exists if a number is a perfect square.
  • The square root of 4 is 2 and the square root of 16 is 4. So we can conclude that the square root of an even perfect square is even.
  • The square root of 9 is 3 and the square root of 81 is 9. So we can conclude that the square root of an odd perfect square is odd.
  • A perfect square cannot be negative and hence the square root of a negative number is not defined.
  • From the above point, it can be concluded that numbers ending with (having unit’s digit) 1, 4, 5, 6, or 9 will have a square root.
  • If a number of ends with an even number of zeros (0’s), then it can have a square root.
  • If the unit digit of a number is 2, 3, 7, or 8 then a perfect square root is not possible.
  • If a number of ends with 2, 3, 7, or 8 (in the unit digit), then the perfect square root does not exist.
  • The two square root values can be multiplied. For example, √5 can be multiplied by √2, then the result should be √10.
  • Two same square roots are multiplied to give a non-square root number. When √5 is multiplied by √5 we get 5 as a result.

Perfect Square 

A number that can be expressed as the product of two identical integers is called a perfect square. Perfect squares are numbers that can be made by squaring any whole number.

9 is a perfect square because it is the product of two equal integers, 3 × 3 = 9. 

However, 10 is not a perfect square because it cannot be expressed as the product of two equal integers. (5 × 2 = 10). 

Thus, a perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself.

The numbers that are perfect squares are mentioned below, and finding the square roots of those numbers is easy. Here are few examples of square roots:

  • 12 = 1
  • 22 = 4
  • 32 = 9
  • 42 = 16
  • 52 = 25
  • 62 = 36
  • 72 = 49
  • 82 = 64
  • 92 = 81
  • 102 = 100

As a result, the complete squares are 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100. 

Methods to find Square Root of a number

To determine if a given number is a perfect square or an imperfect square, we must first determine if it is a perfect square or an imperfect square. If the number is a perfect square, such as 4, 9, 16, etc., we will use the prime factorization process to factorize it. We must use the long division approach to find the root if the number is an incomplete square, such as 2, 3, 5, and so on.

  1. Repeated Subtraction Method
  2. Prime Factorization Method
  3. Division Method

1.  Repeated Subtraction Method 

Example 1: Determine the square root of 25 using the repeated subtraction method.
Solution:

Since, 25 is an odd number. Therefore, the steps to find the square root of 25 is:

  • 25 – 1 = 24
  • 24 – 3 = 21
  • 21 – 5 = 16
  • 16- 7 = 9
  • 9 – 9 = 0

Here it takes five steps to get the 0. 

Hence, the square root of 25 is 5.

Example 2: Determine the square root of 16 using the repeated subtraction method.
Solution:

Since, 16 is an even number. Therefore, the steps to find the square root of 16 is:

  • 16 – 4 = 12
  • 12 – 4 = 8
  • 8 – 4 = 4

Here it takes four steps to get the 0. 

Hence, the square root of 16 is 4.

Example 3: Find the square root of 49 using the repeated subtraction method.

Since, 49 is an odd number. Therefore, the steps to find the square root of 49 is:

  • 49 – 1 = 48
  • 48 – 3 = 45
  • 45 – 5 = 40
  • 40 – 7 = 33
  • 33 – 9 = 24
  • 24 – 11 = 13
  • 13 -13 = 0

Here it takes seven steps to get the 0. 

Hence, the square root of 49 is 7.

2.  Prime Factorization Method 

The prime factorization method involves expressing numbers as a function of their prime factors. The square root of the number is given by the product of one element from each pair of equal prime factors. This approach can also be used to determine whether a given number is a perfect square or not. This method, however, cannot be used to find the square root of non-perfect square decimal numbers.

e.g.: The prime factors of 126 will be 2, 3 and 7 as 2 × 3 × 3 × 7 = 126 and 2, 3, 7 are prime numbers.

  • 16 = 2 × 2 × 2 × 2 = 22 × 22 = √16 = 4
  • 25 = 5 × 5 = 52 = √25 = 5
  • 64 = 2 × 2 × 2 × 2 × 2 × 2 = √64 = 8

3.  Division Method 

When the integers are sufficiently large, it is easy to obtain the square root of a perfect square by utilizing the long division approach, because getting their square roots through factorization becomes lengthy and complicated. To overcome this problem, a new method for finding the square root is developed. This method basically uses the division operation by a divisor whose square is either less than or equal to the dividend.

Step 1: Take the number whose square root is to find. Place a bar over every pair of the digit of the number starting from that in the unit’s place (rightmost side).

Step 2: Let’s divide the leftmost number by the largest number whose square is less than or equal to the number under the leftmost bar. Take this number as the divisor and the quotient. The number under the leftmost bar is considered to be the dividend.

Step 3: Divide and get the number. Bring down the number under the next bar to the right of the remainder.

Step 4: Double the divisor (or add the divisor to itself). To the right of this divisor find a suitable number which together with the divisor forms a new divisor for the new dividend. The new number in the quotient will have the same number as selected in the divisor. The condition is the same as being either less or equal to that of the dividend.

Step 5: Continue this process till we get zero as the remainder. The quotient thus obtained will be the square root of the number.

Example 1: Find the square root of 144 using the division method.

The steps to determine the square root of 144 are:

Step 1: Start the division from the leftmost side. Here 1 is the number whose square is 1. 

Step 2: Putting it in the divisor and the quotient and then doubling it will give as,

Square Root Calculator .Co

Step 3: Now it is required to find a number for the blanks in divisor and quotient. Let that number be x. 

Step 4: Therefore, check when 2x multiplies by x give a number of less than or equal to 44. Take x = 1, 2, 3, and so on and check.

In this case,

  • 21 × 1 = 21
  • 22 × 2 = 44

So we choose x = 2 as the new digit to be put in the divisor and in the quotient. 

The remainder here is 0 and hence 12 is the square root of 144.

Example 2: Find the square root of 196 using the division method.

The steps to determine the square root of 196 are:

Step 1: Start the division from the leftmost side. Here 1 is the number whose square is 1. 

Step 2: Putting it in the divisor and the quotient and then doubling it will give.

Square Root Calculator .Co

Step 3: Now we need to find a number for the blanks in divisor and quotient. Let that number be x. 

Step 4: We need to check when 2x multiplies by x give a number less than or equal to 96. Take x = 1, 2, 3 and so on and check.

In this case, 

  • 21 × 1 = 21
  • 22 × 2 = 44
  • 23 × 3 = 69
  • 24 × 4 = 96

So, choose x = 4  as the new digit to be put in divisor and in the quotient. 

The remainder here is 0 and hence 14 is the square root of 196.

Example 2: Find the square root of 225 using the division method.

The steps to determine the square root of 225 are:

Step 1: Start the division from the leftmost side. Here 1 is the number whose square is 1. 

Step 2: Putting it in the divisor and the quotient and then doubling it will give.

Square Root Calculator .Co

Step 3: Now we need to find a number for the blanks in divisor and quotient. Let that number be x. 

Step 4: We need to check when 2x multiplies by x gives a number which is either less than or equal to 125. Take x = 1, 2, 3 and so on and check.

In this case,

  • 21 × 1 = 21
  • 22 × 2 = 44
  • 23 × 3 = 69
  • 24 × 4 = 96
  • 25 × 5 = 125

So we choose x = 5 as the new digit to be put in divisor and in the quotient. 

The remainder here is 0 and hence 15 is the square root of 225.

4.  Square Roots of Complex Numbers 

To calculate the square root of a complex number, let’s suppose that the root is ea + ib. Then compare it to the original number to get the values of a and b, yielding the square root.

\sqrt{a+ib} = \pm(\sqrt{\frac{\sqrt{a^2+b^2}+a}{2}}+i\frac{b}{|b|}(\sqrt{\frac{\sqrt{a^2+b^2}-a}{2}}))

Example 1: Find the square root of 6 – 8i.

\sqrt{a+ib} = \pm(\sqrt{\frac{\sqrt{a^2+b^2}+a}{2}}+i\frac{b}{|b|}(\sqrt{\frac{\sqrt{a^2+b^2}-a}{2}}))

For the given case, substitute a = 6 and b = (-8) in the above formula,

\begin{aligned}\sqrt{6-8i}&=\pm\sqrt{\dfrac{\sqrt{6^2+(-8)^2}+6}{2}}+i\dfrac{-8}{|-8|}\sqrt{\dfrac{\sqrt{6^2+(-8)^2}-6}{2}}\\&=\pm\sqrt{\dfrac{\sqrt{100}+6}{2}}-i\left(\dfrac{8}{8}\right)\sqrt{\dfrac{\sqrt{100}-6}{2}}\\&=\pm\sqrt{\dfrac{10+6}{2}}-i\sqrt{\dfrac{10-6}{2}}\\&=\pm\left(\sqrt{8}-i(\sqrt{2}\right)\\&=\pm(2\sqrt{2}-i\sqrt{2})\end{aligned}

which is the required solution.

Example 2: Find the square root of 9 + 40i.

\sqrt{a+ib} = \pm(\sqrt{\frac{\sqrt{a^2+b^2}+a}{2}}+i\frac{b}{|b|}(\sqrt{\frac{\sqrt{a^2+b^2}-a}{2}}))

For the given case, substitute a = 9 and b = 40 in the above formula,

\begin{aligned}\sqrt{9+40i}&=\pm\left(\sqrt{\dfrac{\sqrt{9^2+{40}^2}+9}{2}}+i\dfrac{40}{|40|}\sqrt{\dfrac{\sqrt{9^2+(40)^2}-9}{2}}\right)\\&=\pm\left(\sqrt{\dfrac{\sqrt{81+{1600}}+9}{2}}+i\left(\dfrac{40}{40}\right)\sqrt{\dfrac{\sqrt{81+1600}-9}{2}}\right)\\&=\pm\left(\sqrt{\dfrac{\sqrt{1681}+9}{2}}+i\sqrt{\dfrac{\sqrt{1681}-9}{2}}\right)\\&=\pm\left(\sqrt{\dfrac{50}{2}}+i\sqrt{\dfrac{32}{2}}\right)\\&=\pm(5+4i)\end{aligned}

which is the required solution.

Example 3: Find the square root of 3 + 4i.

\sqrt{a+ib} = \pm(\sqrt{\frac{\sqrt{a^2+b^2}+a}{2}}+i\frac{b}{|b|}(\sqrt{\frac{\sqrt{a^2+b^2}-a}{2}}))

For the given case, substitute a = 3 and b = 4 in the above formula,

\begin{aligned}\sqrt{3+4i}&=\pm\left(\sqrt{\dfrac{\sqrt{3^2+{4}^2}+3}{2}}+i\dfrac{4}{|4|}\sqrt{\dfrac{\sqrt{3^2+4^2}-3}{2}}\right)\\&=\pm\left(\sqrt{\dfrac{\sqrt{9+16}+3}{2}}+i\left(\dfrac{4}{4}\right)\sqrt{\dfrac{\sqrt{9+16}-3}{2}}\right)\\&=\pm\left(\sqrt{\dfrac{\sqrt{25}+3}{2}}+i\sqrt{\dfrac{\sqrt{25}-3}{2}}\right)\\&=\pm\left(\sqrt{\dfrac{5+3}{2}}+i\sqrt{\dfrac{5-3}{2}}\right)\\&=\pm(\sqrt{4}+i\sqrt{1})\\&=\pm(2+i)\end{aligned}

which is the required solution.

Does the calculator support fractions?

Yes, simply enter the fraction as a decimal number (use dot as a separator) and you will get the corresponding root. For example, to compute the square root of 1/4 simply enter 0.25 in the number field, press «Calculate» and you will get 0.50 as ouput. If you are having difficulty converting a fraction to a decimal number, you will find our fraction to decimal converter handy.

Square Root of 1 Solved Examples

  1. Find out the perfect square numbers between 1 to 5. Are the numbers rational or irrational? 

    The square root of 1 = 1
    The square root of 2 = 1.414
    The square root of 3 =  1.732
    The square root of 4 = 2
    The square root of 5 = 2.236
    The perfect square numbers between 1 to 5 are 1 and 4.
    The square root of 1 and 4 is a rational number.

  2. Example 2 Jacob is trying different methods to find the square root of a number. Help him find √1 by prime factorization method.

    Prime factorization of 1 = 1 × 1
    Now, the square root of 1 would be the product of 1 digit from each pair.
    Therefore,  √1 = 1

  3. Example If the surface area of a sphere is 4π in2. Find the radius of the sphere.

    Let ‘r’ be the radius of the sphere.
    ⇒ Area of the sphere = 4πr2 = 4π in2
    ⇒ r = ±√1 in
    Since radius can’t be negative,
    ⇒ r = √1
    The square root of 1 is 1.
    ⇒ r = 1 in

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How to Find the Square of a Number?

The square of a number can be found by multiplying a number by itself. For single-digit numbers, we can use multiplication tables to find the square, while in the case of two or more than two-digit numbers, we perform multiplication of the number by itself to get the answer. For example, 9 × 9 = 81, where 81 is the square of 9. Similarly, 3 × 3 = 9, where 9 is the square of 3.

squares and square root of a number

The square of a number is written by raising the exponent to 2. For example, the square of 3 is written as 32 and is read as «3 squared». Here are some examples:

  • 42 = 4 × 4 = 16
  • (-6)2 = -6 × -6 = 36
  • (5/3)2 = 5/3 × 5/3 = 25/9

Simplifying Square Root

For fractions, there is also a similar rule: √x/√y = √(x/y). For example: √50/√10 = √(50/10)= √5

Square Root

The square root of a number is the inverse operation of squaring a number. The square of a number is the value that is obtained when we multiply the number by itself, while the square root of a number is obtained by finding a number that when squared gives the original number.

If ‘a’ is the square root of ‘b’, it means that a × a = b. The square of any number is always a positive number, so every number has two square roots, one of a positive value, and one of a negative value. For example, both 2 and -2 are square roots of 4. However, in most places, only the positive value is written as the square root of a number.

How to Find Square Root?

  • Repeated Subtraction Method
  • Prime Factorization Method
  • Estimation Method
  • Long Division Method

It should be noted that the first three methods can be conveniently used for perfect squares, while the fourth method, i.e., the long division method can be used for any number whether it is a perfect square or not.

Repeated Subtraction Method of Square Root

This is a very simple method. We subtract the consecutive odd numbers from the number for which we are finding the square root, till we reach 0. The number of times we subtract is the square root of the given number. This method works only for perfect square numbers. Let us find the square root of 16 using this method.

  1. 16 — 1 = 15
  2. 15 — 3 =12
  3. 12 — 5 = 7
  4. 7- 7 = 0

You can observe that we have subtracted 4 times. Thus,√16 = 4

Square Root by Prime Factorization Method

  • Step 1: Divide the given number into its prime factors.
  • Step 2: Form pairs of factors such that both factors in each pair are equal.
  • Step 3: Take one factor from the pair.
  • Step 4: Find the product of the factors obtained by taking one factor from each pair.
  • Step 5: That product is the square root of the given number.

Let us find the square root of 144 by this method.

Square root of 144 by prime factorization method

This method works when the given number is a perfect square number.

Finding Square Root by Estimation Method

Estimation and approximation refer to a reasonable guess of the actual value to make calculations easier and more realistic. This method helps in estimating and approximating the square root of a given number. Let us use this method to find √15. Find the nearest perfect square number to 15. 9 and 16 are the perfect square numbers nearest to 15. We know that √16 = 4 and √9 = 3. This implies that √15 lies between 3 and 4. Now, we need to see if √15 is closer to 3 or 4. Let us consider 3.5 and 4. Since 3.52 = 12.25 and 42= 16. Thus, √15 lies between 3.5 and 4 and is closer to 4.

Let us find the squares of 3.8 and 3.9. Since 3.82 = 14.44 and 3.92 = 15.21. This implies that √15 lies between 3.8 and 3.9. We can repeat the process and check between 3.85 and 3.9. We can observe that √15 = 3.872.

This is a very long process and time-consuming.

Calculating Square Root by Long Division Method

Long division is a method for dividing large numbers into steps or parts, breaking the division problem into a sequence of easier steps. We can find the exact square root of any given number using this method. Let us understand the process of finding square root by the long division method with an example. Let us find the square root of 180.

  • Step 1: Place a bar over every pair of digits of the number starting from the units’ place (right-most side). We will have two pairs, i.e., 1 and 80
  • Step 2: We divide the left-most number by the largest number whose square is less than or equal to the number in the left-most pair.

Square root by long division method

Step 3: Bring down the number under the next bar to the right of the remainder. Add the last digit of the quotient to the divisor. To the right of the obtained sum, find a suitable number which, together with the result of the sum, forms a new divisor for the new dividend that is carried down.

square root of 180 is calculated by division method

Step 4: The new number in the quotient will have the same number as selected in the divisor. The condition is the same — as being either less than or equal to the dividend.

Step 5: Now, we will continue this process further using a decimal point and adding zeros in pairs to the remainder.

square root of 180 is 13 point 4 with a remainder in the long division

Square root of 180 by long division is found to be 13 point 1 4 6.

What is Square Root?

The square root of a number is that factor of a number which when multiplied by itself gives the original number. Squares and square roots are special exponents. Consider the number 9. When 3 is multiplied by itself, it gives 9 as the product. This can be written as 3 × 3 or 32. Here, the exponent is 2, and we call it a square. Now when the exponent is 1/2, it refers to the square root of the number. For example, √n = n1/2, where n is a positive integer.

Square Root Definition

The square root of a number is the value of power 1/2 of that number. In other words, it is the number whose product by itself gives the original number. It is represented using the symbol ‘√ ‘. The square root symbol is called a radical, whereas the number under the square root symbol is called the radicand.

Commonly used square roots

Table of commonly encountered square roots:

The calculations were performed using this calculator.

What Is the Square Root of 1?

The square root is an inverse mathematical operation of a square. As you are aware, exceptional cases are always there. In mathematics, there are too many exceptions we deal with. Look at the multiplication shown below when 1 is multiplied by 1 we get a new number as 1, 1 is a square root of 1, or square of 1 is also 1. This is an exceptional case of a square root of 1.

square root of 1

Square Root of 1

The square root of 1 is expressed as √1 in the radical form and as (1)½ or (1)0.5 in the exponent form. It is the positive solution of the equation x2 = 1.

  • Square Root of 1: 1
  • Square Root of 1 in exponential form: (1)½ or (1)0.5
  • Square Root of 1 in radical form: √1

Square of a Number

Any number raised to exponent two (y2) is called the square of the base. So, 52 or 25 is referred to as the square of 5, while 82 or 64 is referred to as the square of 8. We can easily find the square of a number by multiplying the number two times. For example, 52 = 5 × 5 = 25, and 82 = 8 × 8 = 64. When we find the square of a whole number, the resultant number is a perfect square. Some of the perfect squares we have are 4, 9, 16, 25, 36, 49, 64, and so on. The square of a number is always a positive number.

Properties and practical application of square roots

Square roots appear frequently in mathematics, geometry and physics. For example, many physical forces measured in quantities or intensities diminish inversely proportional to the square root of the distance. So, gravity between two objects 4 meters apart will be 1/√4 relative to their gravity at 0 meters. The same is true for radar energy waves, radio waves, light and magnetic radiation in general, and sound waves in gases. It is usually referred to as the «inverse-square law».

The square root is key in probability theory and statistics where it defines the fundamental concept of standard deviation.

square root parabolic curve

Plotting the results from the square root function, as calculated using this square root calculator, on a graph reveals that it has the shape of half a parabola.

What is a square root?

The square root of a number answers the question «what number can I multiply by itself to get this number?». It is the reverse of the exponentiation operation with an exponent of 2, so if r2 = x, then we say that «r is the root of x». Finding the root of a number has a special notation called the radical symbol: √. Usually the radical spans over the entire equation for which the root is to be found. It is called a «square» root since multiplying a number by itself is called «squaring» as it is how one finds the area of a square.

For every positive number there are two square roots — one positive and one negative. For example, the square root of 4 is 2, but also -2, since -2 x -2 = 4. The negative root is always equal in value to the positive one, but opposite in sign. You can see examples in the table of common roots below. Most often when talking about «the root of» some number, people refer to the Principal Square Root which is always the positive root. This is the number our square root calculator outputs as well.

In geometrical terms, the square root function maps the area of a square onto its side length. The function √x is continuous for all nonnegative x and differentiable for all positive x.

Square Root of a Negative Number

The square root of a negative number cannot be a real number, since a square is either a positive number or zero. But complex numbers have the solutions to the square root of a negative number. The principal square root of -x is: √(-x)= i√x. Here, i is the square root of -1.

For example: Take a perfect square number like 16. Now, let’s see the square root of -16. There is no real square root of -16. √(-16)= √16 × √(-1) = 4i (as, √(-1)= i), where ‘i’ is represented as the square root of -1. So, 4i is a square root of -16.

How calculate a square root

  1. Start with a guess (b). If a is between two perfect squares, a good guess would be a number between those squares.
  2. Divide a by b: e = a / b. If the estimate e is an integer, stop. Also stop if the estimate has achieved the desired level of decimal precision.
  3. Get a new guess b1 by averaging the result of step #2 e and the initial guess b: b1 = (e + b) / 2
  4. Go to step #2 using b1 in place of b

For example, to find the square root of 30 with a precision of three numbers after the decimal point:

Step 1: a = 30 is between 25 and 36, which have roots of 5 and 6 respectively. Let us start with b = 5.5.
Step 2: e = a / b = 30 / 5.5 = 5.45(45). Since b is not equal to e (5.500 ≠ 5.454), continue calculation.
Step 3: b1 = (5.45 + 5.5) / 2 = 5.47727(27)
Step 4: e = 30 / 5.47727 = 5.477178. Since b1 = e = 5.477 within three position after the decimal point, stop the square root-finding algorithm with a result of √30 = 5.47727(27).

Checking the outcome against the square root calculator output of 5.477226 reveals that the algorithm resulted in a correct solution. While the above process can be fairly tedious especially with larger roots, but will help you find the square root of any number with the desired decimal precision.

Square Root Formula

Square Root Table

The square root table consists of numbers and their square roots. It is useful to find the squares of numbers as well. Here is the list of square roots of perfect square numbers and some non-perfect square numbers from 1 to 10.

The numbers that are not perfect squares are irrational numbers.

How to Find the Square Root of 1?

Explore some square roots of other numbers using illustrations and interactive examples.

  • Square root 5
  • Square root 2
  • Square root 4
  • Square root 11
  • Square root 9
  • Can you think what will be the square root of -11 and -111?
  • 1 is the only number whose square and the square root are equal.
  • 1 is neither prime nor composite.

Is Square Root of 1 Rational or Irrational?

Since √1 = 1 which is rational numbers. Hence, the square root of 1 is rational. 

Squares and Square Roots

There is very strong relation between squares and square roots as each one of them is the inverse relation of the other. i.e., if x2 = y then x = √y. It can be simply remembered like this:

  • When «square» is removed from one side of the equation, we get the square root on the other side. For example, 42 = 16 means, 4 = √16. This is also known as «taking square root on both sides».
  • When «square root» is removed from one side of the equation, we get square on the other side. For example, √25 = 5 means, 25 = 52. This is also known as «squaring on both sides»

Example: Solve the equation √(2x + 3) = 10.

Squaring on both sides of the equation would result in the cancellation of the square root on the left side.

2x + 3 = 102

2x + 3 = 100

2x = 97

x = 97/2 = 48.5

Here are more differences between squares and square roots.

☛ Related Articles

Square Root of Numbers

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FAQs on the Square Root of 1

What is the Value of the Square Root of 1?

The square root of 1 is 1.

If the Square Root of 1 is 1. Find the Value of the Square Root of 0.01.

Let us represent √0.01 in p/q form i.e. √(1/100) = 0.01/10 = 0.1. Hence, the value of √0.01 = 0.1

What is the Square Root of -1?

The square root of -1 is an imaginary number. It is also written as √-1 = i
where i = √-1 and it is called the imaginary unit.

FAQs on Square Root

What is Square Root in Math?

The square root of a number is a number that when multiplied by itself gives the actual number. For example, 2 is the square root of 4, and this is expressed as √4 = 2. This means when 2 is multiplied by 2 it results in 4 and this can be verified as 2 × 2 = 4.

How to Calculate the Square Root of a Number?

It is very easy to find the square root of a number that is a perfect square. For example, 9 is a perfect square, 9 = 3 × 3. So, 3 is the square root of 9 and this can be expressed as √9 = 3. The square root of any number, in general, can be found by using any of the four methods given below:

  • Repeated Subtraction Method
  • Prime Factorization Method
  • Estimation and Approximation Method
  • Long Division Method

Can Square Root be Negative?

Yes, the square root of a number can be negative. In fact, all the perfect squares like 4, 9, 25, 36, etc. have two square roots, one is a positive value and one is a negative value. For example, the square roots of 4 are -2 and 2. To verify this, we can see that (-2) × (-2) = 4. Similarly, the square roots of 9 are 3 and -3.

How to Find the Square Root of a Decimal Number?

The square root of a decimal number can be found by using the estimation method or the long division method. In the case of decimal numbers, we make pairs of whole number parts and fractional parts separately. And then, we carry out the process of long division in the same way as any other whole number.

What is the Square Root Symbol?

The symbol that is used to denote square root is called the radical sign ‘√ ‘. The term written inside the radical sign is called the radicand.

How to Multiply Two Square Root Values Together?

Let us say we have two numbers a and b. First, we will find the square root of the numbers a and b. Then, after finding the square root we will multiply the square roots value together. Let us understand this with a practical illustration. For example, multiply √4 × √16. The square root of 4 is 2 (√4 = 2) and the square root of 16 is 4 (√16 = 4). Now, we will multiply the value of the square root of 4 and 16, i.e., 2 × 4 = 8. Instead, we can apply the property of square roots, √a × √b = √ab.

What is the Formula for Calculating the Square Root of a Number?

The square root of any number can be expressed using the formula: √y = y½. In other words, if a number has 1/2 as its exponent, it means we need to find the square root of the number.

What is the Square and Square Root of a Number?

The square of a number is the product that we get on multiplying a number by itself. For example, 6 × 6 = 36. Here, 36 is the square of 6. The square root of a number is that factor of the number and when it is multiplied by itself the result is the original number. Now, if we want to find the square root of 36, that is, √36, we get the answer as, √36 = 6. Hence, we can see that the square and the square root of a number are inverse operations of each other.

Which Method is Used to Find the Square Root of Non-Perfect Square Numbers?

In Math, a non-perfect or an imperfect square number is considered as a number whose square root cannot be found as an integer or as a fraction of integers. The square root of a non-perfect square number can be calculated by using the long division method.

How to Find a Square Root on a Calculator?

To find the square root value of any number on a calculator, we simply need to type the number for which we want the square root and then insert the square root symbol √ in the calculator. For example, if we need to find the square root of 81, we should type 81 in the calculator and then press the symbol √ to get its square root. We will get √81 = 9.

What are the Applications of the Square Root Formula?

There are various applications of the square root formula:

  • The square root formula is mainly used in algebra and geometry.
  • It helps in finding the roots of a quadratic equation.
  • It is widely used by engineers.

What does the Square of a Number mean?

The product that we get on multiplying a number by itself is the square of the number. For example, 5 × 5 = 25. Here, 25 is the square of 5 and this can also be written as 52 = 25.

How to Calculate the Square Root of a Negative Number?

Note that the square root of a negative number is not a real number. It is an imaginary number. For example, √(-4) = √(-1) × √4 = i (2) = 2i, where ‘i’ is known as «iota» and i2 = -1 (or) i = √(-1).

Why is the Square of a Negative Number Positive?

The square of a negative number is positive because when two negative numbers are multiplied it always results in a positive number. For example, (-4) × (-4) = 16.

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