# What Is Square Root? What Are the Methods Used to Find the Square Root?

## Repeated subtraction method

The repeated subtraction method first subtracts the given number from the odd numbers until you get zero. The step in which zero is obtained is the square root of the number. It is a time-consuming and lengthy process.
Find the square root of 36 using the repeated subtraction method

1. 36-1 = 35
2. 35-3 = 32
3. 32-5 = 27
4. 27-7 = 20
5. 20-9 = 11
6. 11-11 = 0

We got zero in the sixth step. Hence,  √36 = 6.

Square root calculation is used in finance, quadratic formula, standard deviation lengths and distance, and many more. With this blog, we will understand what is square root, the formula and symbol, and some solved examples.

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## What are the methods to find the square root of a number?

The numbers that end with 2,3, 7 0r 8 at the unit place are not perfect square numbers and are a little difficult to find. The numbers ending with 1,4,5,6 or 9 in the unit place are perfect square numbers and can be found easily using the prime factorization or the long division method.
The method to find the square root is

• Prime factorization method
• Long division method
• Repeated subtraction method

## Prime factorization method

Prime factorization is one of the easiest methods to find the square root of any number. Let us take n as the prime number, by grouping the similar numbers we get n2, by multiplying the similar n2. The value we get is the square root of the number.
Let us take the number 144 as an example and solve them using the prime factorization method.
The prime factor of 144 = 12×12
By squaring them we get 122, since two is the only similar square we take the common one.
Here, the common square is 12.
Hence, the √144 = 12

## Long division method

• Step 1: Find the smallest integer that can divide the number.
• Step 2: Keep following the long division using divisor and dividend.
• Step 3: When the particular number of satisfaction is reached the quotient is the square root of the number.

Find the square root of 2

• Step 1: Find the smallest integer that can divide the number. 1 is the perfect root number that is smaller and closest to 2.
• Step 2: Keep following the long division using divisor and dividend.
• Step 3: When the particular number of satisfaction is reached the quotient is the square root of the number. Hence, the √2 = 1.4142

What is square root? A number multiplied by itself is called the square of that number. So $x \times x=x^2$ is the square of $x.$ Here the number $x$ is called the square root of $x^2.$

#### Definition of Square Root

Thus, if $y=x^2$ then we say $x$ is the square root of $y.$

#### Square Root Examples

Few examples of square roots:

#### Rules of Square Root

We list all the properties of square roots.

• $x^{1/2}=\sqrt{x}$
• $\sqrt{x^2}=x$
• $a\sqrt{x}+b\sqrt{x}=(a+b)\sqrt{x}$
• $a\sqrt{x}-b\sqrt{x}=(a-b)\sqrt{x}$
• $\sqrt{x \times y}=\sqrt{x} \times \sqrt{y}$
• $\sqrt{\frac{x}{y}}=\frac{\sqrt{x}}{\sqrt{y}}$   if $y \neq 0$

#### Square Root Table

Square toot table: (perfect square)

#### Square Root as a Function

Note that $f$ is a one-to-one function but not onto.

#### Methods of Finding Square Root

• Square root by prime factorization
• Square root by repeated subtraction method
• Square root by estimation method
• Square root by long division method

#### Simplifying Square Roots

Now, we will learn how to simplify square roots. Let us simplify the square root of $27$.

Firstly, we will factorize $27.$ Note that

$27=3 \times 3 \times 3$

Taking square root on both sides, we get that

#### Addition or Subtraction of Square Roots

First, we express the square roots into their simplified forms as above.

Firstly, we will simplify both square roots.

#### Multiplication or Division of Square Roots

First, we express the square roots into their simplified forms as above.

#### Square Root of a Decimal

At first, we have to express the decimal number as a fraction.

Now we will apply the method of finding the square root of real numbers to compute √x and √y.  And we put their values in the fraction obtained in step 2.

At last, we will simplify the fraction obtained in step 3. The resultant value will be the square root of the given decimal number.

Let’s understand the above process by an example.

Taking square root we get

So the square root of $0.16$ is $0.4$

#### Square Root of a Negative Number

We know that the square of any number is always positive. So by definition, the square root of a negative number does not exist in the set of real numbers. But they do exist in the world of complex numbers.

#### Square Root of a Complex Number

The computation of the square root of a complex number is not as simple as the method of finding the square root of a real number. We know that the general form of a complex number is $a+ib,$ where both $a$ and $b$ are real numbers. The formula for finding the square root of $a+ib$ is given below:

Using the above formula, we can find the square root of $i.$ Note that $a=0$ and $b=1$ as $i=0+i \cdot 1.$ So we have

In a similar way, one can find the square root of $-i$ also.

#### Square Root of a 2×2 matrix

Square root of a matrix: Let $A$ and $B$ be two matrices such that $A^2=B.$ In this case, we say that the matrix $A$ is the square root of $B.$ Note that there may have many square roots of $B,$ for example, $I_2,$ the 2×2 identity matrix, has infinitely many square roots. So it is natural to ask the question: is the square root of a matrix well-defined? The answer is yes. We will discuss it now.

Let $B$ be a positive-definite matrix. We know that there is only one positive-definite matrix $A$ such that $A^2=B.$ So we define

The square root of $B$ is well-defined. We note that among the infinite number of square roots of $I_2,$ there is exactly one positive-definite which we define to be the square root of $I_2.$

#### Generalization of Square Roots

Let $f(x)$ be a polynomial. If a number $c$ satisfies the equation $f(c)=0,$ then $c$ is called a root of $f(x).$ This type of roots are known as polynomial roots.

#### Square Root Applications

The computation of square roots has many applications in several branches of mathematics; such as

• Equation solving
• Polynomial
• Ring theory, field theory
• Numerical analysis
• Geometry (for example, to find the side of a square if the area is known)

#### Solved Problems of Square Roots

Find the square root of $125.$

Note that $125=5 \times 5 \times 5$

#### How to Find Square Root of Numbers

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.

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## 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.

$36 – 1 = 35$

$35 – 3 = 32$

$32 – 5 = 27$

$27 – 7 = 20$

$20 – 9 = 11$

$11 – 11 = 0$

### 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.

## 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

### 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 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.

## Properties of Square Root Function

Here are the important points/properties that are to be noted about the square root function f(x) = √x.

• Its domain is [0, ∞).
• Its range is [0, ∞).
• It has no relative maxima but it has a minimum at (0, 0).
• A square root function has no asymptotes.
• It is an increasing function throughout its domain [0, ∞).
• The square root function f(x) = √x has critical point at (0, 0) and it has no inflection points.

## Perfect square and not a perfect square

A perfect square number is one whose square roots give a whole number. The square root of 4 is 2, the square root of 400 is 20, etc.,
A not-perfect square number is one whose roots are not whole numbers but rather decimals.
The square root of 5 = 7.071

## 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.

## 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

## 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.

## Graphing Any Square Root Function

We have seen how to graph the parent square root function f(x) = √x. Here are the steps that are useful in graphing any square root function that is of the form f(x) = a√(b(x — h)) + k in general.

• Step 1: Identify the domain of the function by setting «the expression inside the square root» to greater than or equal to 0 and solving for x.
• Step 2: The range of any square root function is always y ≥ k where ‘k’ is the vertical translation of the function f(x) = a√(b(x — h)) + k.
• Step 3: Construct a table of values with two columns x and y, take some random numbers for x (from the domain only) starting from the first value of the domain, substitute them in the given function and find the corresponding values of y.
• Step 4: Plot all the points on the plane and connect them by a curve and also extend the curve following the same trend.

Note: Computing the x-intercept and y-intercept would also help in graphing the square root function.

Example: Graph the square root function f(x) = √(x — 2) + 3.

To find its domain, x — 2 ≥ 0 ⇒ x ≥ 2.

Its vertical shift is 3 and hence its range is y ≥ 3.

Now, we will construct a table with some values greater than 2 (as the domain is x ≥ 2). Choose some values for x such that √(x — 2) is a perfect square so that the calculation becomes easier.

Now, plot these points and join them by a curve.

We can also graph the square root function by applying the transformations on the parent square root graph f(x) = √x.

## The formula for finding the square root

The formula is simple x = √y or x = y2

## What is Square Root Function?

f(x) = √x is the parent square root function but when the transformations are applied to it, it may look like f(x) = a√(b(x — h)) + k, where a, b, h, and k are numbers such that

• ‘a’ is the vertical dilation
• ‘b’ is the horizontal dilation
• ‘h’ is the horizontal translation
• ‘k’ is the vertical translation

## Square root

The square root of a value is a number when multiplied by itself gives the original number or the square root is the inverse of squaring the numbers. Which tells that both concepts are dependable on each other.

The square root of a number in the radical form is denoted as √ when they are in the exponential form they are denoted as the (number)1/2 or the (number) 0.5.

Let us take p and q. The square root of p = √q or p = q2 the number that is positive when positive or multiplied gives the exact number.
That is if we take the number 9 the square root of 9 is found by multiplying the number twice.
9 = 32 or √9 = 3, when the number three is multiplied it gives the number 9.

## 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.

## 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.

## 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.

## Square Root Function

The square root function involves the square root symbol √ (which is read as «square root of»). The square root of a number ‘x’ is a number ‘y’ such that y2 = x. i.e., if y2 = x ⇒ y = √x. i.e., if ‘x’ is the square of ‘y’ then ‘y’ is the square root of ‘x’. Some examples are:

• 22 = 4 ⇒ √4 = 2
• 42 = 16 ⇒ √16 = 4

We know that the square root of a number can be either positive or negative. i.e.,√4 = ±2. But while defining the square root function, we restrict its range to be the set of all positive real numbers (otherwise it won’t become a function at all), and hence in the case of the square root function, the result is always positive. Let us use all these facts to understand the square root function.

## Solved examples

Q1: Solve the equation √2a+9 = 5
A1: √2a+9 = 5
Let us take square on both sides
2a+9 = 52
2a+9 = 25
2a = 25-9
2a = 16
a = 16/2
a=8

Q2: Square root of 125 using the prime factorization method
A2: The prime factors of 125 = 5×5×5
√125 = 5×5
= 5 × 2.236
= 11.8

Q3: Square root of 12 using the long division method

A 3: √12 = 3.464

Q4: Find the square root of pi

A 4: √𝛑 = 1.77

Q5: Find the square root of 64 using the repeated subtraction method

1. 64-1 = 63
2. 63-3 = 60
3. 60 – 5 = 55
4. 55-7 = 48
5. 48 – 9 = 39
6. 39-11 = 28
7. 28-13 = 15
8. 15-15 = 0
9. √64 = 8

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## FAQs on Square Root Function

### What is Parent Square Root Function?

The parent square root function is f(x) = √x. This function may be translated/dilated/reflected and can transform to the form f(x) = a√(b(x — h)) + k.

### What is the Formula of Square Root Function?

The formula for the square root function is f(x) = √x. It means the output of each input value is equal to the square root of the input value. For example, f(25) = √25 = 5. Note that all inputs and outputs of a square root function are always non-negative.

### What is the Derivative of Square Root Function?

The derivative of the square root function f(x) = √x is calculated by the power rule of differentiation, d(xn)/dx = nxn-1. By this rule, d(√x)/dx = d(x1/2)/dx = (1/2) x(1/2) — 1 = (1/2) x-1/2 = 1/(2√x). Thus, the derivative of √x is 1/(2√x).

### How to Graph Square Root of x?

To graph the square root of x, just note that its inputs and outputs are all non-negative and hence its graph lies in the first quadrant. Further, to get the clear shape of the graph, calculate some points on it, by taking some random numbers for x and computing corresponding y-values for them.

### What is the Difference Between Cube Root Graph and Square Root Graph?

The cube root graph can take in any real number as input and produces any real number as output. But the square root function takes in and produces only the non-negative real numbers.

### What are the Asymptotes of Square Root Function?

There is no horizontal line or vertical line that can break the graph of square root function and hence it has no vertical/horizontal asymptotes.

### What is the Integral of Square Root Function?

The integral of the square root function √x can be found using the power rule of integration ∫xn dx = xn+1/(n + 1) + C.. Using this, ∫√x dx = ∫x1/2 dx = x(1/2 + 1)/(1/2 + 1) + C = x3/2/(3/2) + C = (2/3) x3/2 + C.

## Square Root Formula

What is a square root?
The square root of a value is a number when multiplied by itself gives the original number or the square root is the inverse of squaring the numbers. Which tells that both concepts are dependable on each other

Give an example of a perfect square number.
The numbers that are perfect squares are 4,16 and 25 are some of the examples for perfect square numbers.

What are irrational numbers?
Any number that is not expressed as a fraction is called an irrational number.

What is the square root of -1?
We know the definition of square root, thus multiplying or squaring these values we get 1 and not a negative result.
-1 = (-1) (-1) = 1. The square root of -1 is i

The square root symbol is denoted as √ called a radicand.

What is the formula for square root?
The formula is simple x = y or x = y2

What is the square root of 100?
The square root of 100 is 10

## Symbol of the square root

The square root symbol is denoted in a radical way. Let us say we have to express x in the square root then it is denoted as x. The symbol in radical form is also called radicant. Here x = any number.
The square root of 4 is expressed as √4

## 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.

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

## Square Root of Numbers

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## The square root of -1

Understanding the square root of minus values is simple. There is no real square root for the negative integers which are often referred to as imaginary roots. Why? Because what happens when you multiply two negative values is you get a positive number and not a negative one. Let’s say we need to find the square root of -1. We know the definition of square root, thus multiplying or squaring these values we get 1 and not a negative result.
-1 = (-1) (-1) = 1.
Therefore they are referred to as imaginary roots and are denoted as I.
Find the square root of -2
√-2 = 1.4142i

## Rational and irrational numbers

Rational numbers are expressed as a fraction of the integer. That is they are simple fractions and p/q ≠0.
Let us take the number 121 as an example:
The square root of 121 = 11, when we divide the number 121/11 = 11 and not zero or a complicated fraction.
An irrational number leads to complicated fractions and is often difficult to solve. Any number that is not expressed as a fraction is called an irrational number.
The square root of 5 = 7.071 which cannot be further converted into a fraction and is also not equal to 0.

## Domain and Range of Square Root Function

The square root of a negative number is NOT a real number. i.e., the square root function cannot accept negative numbers as inputs. i.e.,

• The domain of the square root function f(x) = √x is the set of all non-negative real numbers. i.e., the square root function domain is [0, ∞). Note that it includes 0 as well in the domain.

In general, the square root of a number can be either positive or negative. i.e., √25 = 5 or -5 as 52 = 25 and (-5)2 = 25. But the range of the square root function (i.e., its y-values) is restricted to only positive numbers, because otherwise, it fails the vertical line test and it won’t be a function if an input has two outputs. Thus,

• The range of the square root function f(x) = √x is also the same as its domain [0, ∞). Note that it includes 0 as well in the range.

## Square Root Graph

Note that when some transformations are applied to the graph, the graph may not lie in the first quadrant itself.

## 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 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.

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.

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.

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.

## Simplifying Square Root

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

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