Square root of 23.5

Square Root Calculator

Square Root calculator enumerates the square root of a given number. Square root is defined as a value that can be multiplied by itself to get the original number. The square of a number x will be a number y that satisfies the equation y² = x.

What is a Square Root Calculator?

Square Root Calculator is an online tool used to calculate the square root for a given number that is a positive integer. The square root of a number can also be expressed as the number raised to the power 1/2. To use this square root calculator, enter values in the input box.

Square Root Calculator

NOTE: Enter number upto 3 digits only.

How to Use the Square Root Calculator?

• Step 1: Go to online square root calculator.
• Step 2: Enter a positive number in the input box of the square root calculator.
• Step 3: Click on the «Calculate» button to find the square root of the number.
• Step 4: Click on the «Reset» button to clear the fields and enter new values.

How Does Square Root Calculator Work?

1. Repeated subtraction method.
2. Prime Factorization method.
3. Estimation Method.
4. Long Division Method.

The first three methods are very useful in determining the square root of a perfect square. A positive number that can be written as the multiplication of the number by itself can be defined as a perfect square. The long division method is more laborious however, it can be used in finding the square root of any number. Such a number does not have to be a perfect square. If the unit’s place of a number has 2, 3, 7, or 8 then the perfect square root will not exist. However, if the unit’s place of a number has 1, 4, 5, 6, or 9 then such a number can have a perfect square root. The formula for a square root is given as

y = √x

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Use our free online calculator to solve challenging questions. With Cuemath, find solutions in simple and easy steps.

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Solved Examples on Square Root Calculator

Find the square root of 64 and verify it using the square root calculator.

Using the Prime Factorization Method

64 = 2 × 2 × 2 × 2 × 2 × 2

64 = 2² × 2² × 2²

64 = (2 × 2 × 2)²

64 = 8²

√64 = 8

Therefore, 8 is the square root 64.

Find the square root of 25 and verify it using the square root calculator.

Using the Prime Factorization Method

25 = 5 × 5

25 = 5²

√25 = 5

Therefore, 5 is the square root 25.

• 800
• 216

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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.

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 = n^1/2, 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.

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.

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 Formula

Simplifying Square Root

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

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.

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.

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

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.

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Square Root of Numbers

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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 = y^½. 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 Fraction Calculator

Square Root Fraction Calculator is an online tool that helps to calculate the square root of a given fraction. 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.

What is Square Root Fraction Calculator?

Online Square Root Fraction Calculator helps you to calculate the square root of a given fraction in a few seconds.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.

Square Root Fraction Calculator

NOTE: Enter the input values upto 3 digits.

How to Use the Square Root Fraction Calculator?

• Step 1: Enter the numerator and denominator in the respective input box.
• Step 2: Click on the «Calculate» button to find the square root fractions.
• Step 3: Click on the «Reset» button to clear the fields and enter the new values.

How to Find Square Root Fractions?

The square root of a number is defined as a value, which gives the number when it is multiplied by itself, the symbol √  is used to represent a square root function. If the given number is not a  perfect square fraction then we can always use the long division method to find the same.

The square root of fraction equation is given by f(x) = √x

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Use our free online calculator to solve challenging questions. With Cuemath, find solutions in simple and easy steps.

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Solved Examples on Square Root Fraction Calculator

Find the Square root fraction of 20/29

f(x) = √x

f(20 / 29) = √ 20 / 29

= √20 / 29

Find the Square root fraction of 16/49

f(x) = √x

f(16 / 49) = √ 16 / 49

Find the Square root fraction of 81/24

f(x) = √x

f(81/24) = √ 81 / 24

= √81 / √24

Similarly, you can try the square root fraction calculator to find square root fractions calculator for

• f = 20/105

• f = 34/65

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In this article we’re going to calculate the square root of 22 and explore what the square root is and answer some of the common questions you might. We’ll also look at the different methods for calculating the square root of 22 (both with and without a computer/calculator).

Square Root of 22 Definition

In mathematical form we can show the square root of 22 using the radical sign, like this: √22. This is usually referred to as the square root of 22 in radical form.

Want to quickly learn or refresh memory on how to calculate square root play this quick and informative video now!

So what is the square root? In this case, the square root of 22 is the quantity (which we will call q) that when multiplied by itself, will equal 22.

√22 = q × q = q²

Is 22 a Perfect Square?

In math, we refer to 22 being a perfect square if the square root of 22 is a whole number.

In this case, as we will see in the calculations below, we can see that 22 is not a perfect square.

To find out more about perfect squares, you can read about them and look at a list of 1000 of them in our What is a Perfect Square? article.

Is The Square Root of 22 Rational or Irrational?

A common question is to ask whether the square root of 22 is rational or irrational. Rational numbers can be written as a fraction and irrational numbers cannot.

A quick way to check this is to see if 22 is a perfect square. If it is, then it is a rational number. If it’s not a perfect square then it’s an irrational number.

We already know if 22 is a perfect square so we also can see that √22 is an irrational number.

Can the Square Root of 22 Be Simplified?

22 can be simplified only if you can make 22 inside the radical symbol smaller. This is a process that is called simplifying the surd. In this example square root of 22 cannot be simplified.

√22 is already in its simplest radical form.

How to Calculate The Square Root of 22 with a Calculator

√22 ≈ 4.6904

How to Calculate the Square Root of 22 with a Computer

On a computer you can also calculate the square root of 22 using Excel, Numbers, or Google Sheets and the SQRT function, like so:

SQRT(22) ≈ 4.690415759823

What is the Square Root of 22 Rounded?

Sometimes you might need to round the square root of 22 down to a certain number of decimal places. Here are the solutions to that, if needed.

10th: √22 ≈ 4.7

100th: √22 ≈ 4.69

1000th: √22 ≈ 4.69

What is the Square Root of 22 as a Fraction?

We covered earlier in this article that only a rational number can be written as a fraction, and irrational numbers cannot.

Like we said above, since the square root of 22 is an irrational number, we cannot make it into an exact fraction. However, we can make it into an approximate fraction using the square root of 22 rounded to the nearest hundredth.

≈ 4 34.5/50

What is the Square Root of 22 Written with an Exponent?

All square root calculations can be converted to a number (called the base) with a fractional exponent. Let’s see how to do that with the square root of 22:

√b = b^½

√22 = 22^½

How to Find the Square Root of 22 Using Long Division

Finally, we can use the long division method to calculate the square root of 22. This is very useful for long division test problems and was how mathematicians would calculate the square root of a number before calculators and computers were invented.

Step 1

Set up 22 in pairs of two digits from right to left and attach one set of 00 because we want one decimal:

Step 2

Starting with the first set: the largest perfect square less than or equal to 22 is 16, and the square root of 16 is 4 . Therefore, put 4 on top and 16 at the bottom like this:

Step 3

Calculate 22 minus 16 and put the difference below. Then move down the next set of numbers.

Step 4

Double the number in green on top: 4 × 2 = 8. Then, use 8 and the bottom number to make this problem:

8? × ? ≤ 600

The question marks are «blank» and the same «blank». With trial and error, we found the largest number «blank» can be is 6.

Now, enter 6 on top:

That’s it! The answer shown at the top in green. The square root of 22 with one digit decimal accuracy is 4.7.
Notice that the last two steps actually repeat the previous two. To add decimal places to your answe you can simply add more sets of 00 and repeat the last two steps.

In this article we’re going to calculate the square root of 17 and explore what the square root is and answer some of the common questions you might. We’ll also look at the different methods for calculating the square root of 17 (both with and without a computer/calculator).

Square Root of 17 Definition

In mathematical form we can show the square root of 17 using the radical sign, like this: √17. This is usually referred to as the square root of 17 in radical form.

Want to quickly learn or refresh memory on how to calculate square root play this quick and informative video now!

So what is the square root? In this case, the square root of 17 is the quantity (which we will call q) that when multiplied by itself, will equal 17.

√17 = q × q = q²

Is 17 a Perfect Square?

In math, we refer to 17 being a perfect square if the square root of 17 is a whole number.

In this case, as we will see in the calculations below, we can see that 17 is not a perfect square.

To find out more about perfect squares, you can read about them and look at a list of 1000 of them in our What is a Perfect Square? article.

Is The Square Root of 17 Rational or Irrational?

A common question is to ask whether the square root of 17 is rational or irrational. Rational numbers can be written as a fraction and irrational numbers cannot.

A quick way to check this is to see if 17 is a perfect square. If it is, then it is a rational number. If it’s not a perfect square then it’s an irrational number.

We already know if 17 is a perfect square so we also can see that √17 is an irrational number.

Can the Square Root of 17 Be Simplified?

17 can be simplified only if you can make 17 inside the radical symbol smaller. This is a process that is called simplifying the surd. In this example square root of 17 cannot be simplified.

√17 is already in its simplest radical form.

How to Calculate The Square Root of 17 with a Calculator

√17 ≈ 4.1231

How to Calculate the Square Root of 17 with a Computer

On a computer you can also calculate the square root of 17 using Excel, Numbers, or Google Sheets and the SQRT function, like so:

SQRT(17) ≈ 4.123105625618

What is the Square Root of 17 Rounded?

Sometimes you might need to round the square root of 17 down to a certain number of decimal places. Here are the solutions to that, if needed.

10th: √17 ≈ 4.1

100th: √17 ≈ 4.12

1000th: √17 ≈ 4.123

What is the Square Root of 17 as a Fraction?

We covered earlier in this article that only a rational number can be written as a fraction, and irrational numbers cannot.

Like we said above, since the square root of 17 is an irrational number, we cannot make it into an exact fraction. However, we can make it into an approximate fraction using the square root of 17 rounded to the nearest hundredth.

≈ 4 3/25

What is the Square Root of 17 Written with an Exponent?

All square root calculations can be converted to a number (called the base) with a fractional exponent. Let’s see how to do that with the square root of 17:

√b = b^½

√17 = 17^½

How to Find the Square Root of 17 Using Long Division

Finally, we can use the long division method to calculate the square root of 17. This is very useful for long division test problems and was how mathematicians would calculate the square root of a number before calculators and computers were invented.

Step 1

Set up 17 in pairs of two digits from right to left and attach one set of 00 because we want one decimal:

Step 2

Starting with the first set: the largest perfect square less than or equal to 17 is 16, and the square root of 16 is 4 . Therefore, put 4 on top and 16 at the bottom like this:

Step 3

Calculate 17 minus 16 and put the difference below. Then move down the next set of numbers.

Step 4

Double the number in green on top: 4 × 2 = 8. Then, use 8 and the bottom number to make this problem:

8? × ? ≤ 100

The question marks are «blank» and the same «blank». With trial and error, we found the largest number «blank» can be is 1.

Now, enter 1 on top:

That’s it! The answer shown at the top in green. The square root of 17 with one digit decimal accuracy is 4.1.
Notice that the last two steps actually repeat the previous two. To add decimal places to your answe you can simply add more sets of 00 and repeat the last two steps.

Use the square root calculator below to find the square root of any imaginary or real number. See also in this web page a Square Root Table from 1 to 100 as well as the Babylonian Method or Hero’s Method.

The Babylonian Method also known as Hero’s Method

See below how to calculate the square root of 81 step-by-step using the Babylonian Method also known as Hero’s Method.

What is square root?

Definition of square root

A square root of a number ‘a’ is a number x such that x² = a, in other words, a number x whose square is a. For example, 9 is the square root of 81 because 9² = 9•9 = 81, -9 is square root of 81 because (-9)² = (-9)•(-9) = 81.

Square Root Table 1-100

Square roots from 1 to 100 rounded to the nearest thousandth.

Use the square root calculator below to find the square root of any imaginary or real number. See also in this web page a Square Root Table from 1 to 100 as well as the Babylonian Method or Hero’s Method.

The Babylonian Method also known as Hero’s Method

See below how to calculate the square root of 23.5 step-by-step using the Babylonian Method also known as Hero’s Method.

What is square root?

Definition of square root

A square root of a number ‘a’ is a number x such that x² = a, in other words, a number x whose square is a. For example, 4 is the square root of 16 because 4² = 4•4 = 16, -4 is square root of 16 because (-4)² = (-4)•(-4) = 16.

Square Root Table 1-100

Square roots from 1 to 100 rounded to the nearest thousandth.

Use the square root calculator below to find the square root of any imaginary or real number. See also in this web page a Square Root Table from 1 to 100 as well as the Babylonian Method or Hero’s Method.

The Babylonian Method also known as Hero’s Method

See below how to calculate the square root of 122 step-by-step using the Babylonian Method also known as Hero’s Method.

What is square root?

Definition of square root

A square root of a number ‘a’ is a number x such that x² = a, in other words, a number x whose square is a. For example, 11 is the square root of 121 because 11² = 11•11 = 121, -11 is square root of 121 because (-11)² = (-11)•(-11) = 121.

Square Root Table 1-100

Square roots from 1 to 100 rounded to the nearest thousandth.

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

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.

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.

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.

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.

Commonly used square roots

Table of commonly encountered square roots:

The calculations were performed using this calculator.

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.

Do you want to learn how to solve some square root math problems? Here are some examples of square root problems that we have explained and solved:

Unit digit of the square of 272

Square Root of Pi

10000 square feet to acres

What is the square root of 3?

What is the square root of 5 squared?

What is the square root of 1/4?

What is sum of the first 10 square numbers?

Is the square root of 40 rational or irrational?

Simplify Square Root of 72

How to find the square root of 80 by hand?

What is the square root of 50 as a fraction?

Where can I get a list of perfect squares?

What is the square root of negative 4?

How to convert 200 square meters to square feet

Convert 2 acres to square feet

How many square feet is a 12×12 room?

What is Square Root of 0.25?

1 square foot to square meters

Here we will define, explain, and show you how to calculate the square root of Pi. First, note that Pi or Pie as it is sometimes
referred to can be written with the Pi symbol: π

Pi (π) is not a perfect square, but an irrational decimal number that cannot be simplified. Pi goes
on indefinitely, so obviously it is not possible to list the entire Pi. However, below is the beginning of Pi.

Square root of Pi definition
The square root of Pi in mathematical form is written with the radical sign like this: √π. We call this the
square root of Pi in radical form.
The square root of Pi is a quantity (q) that when multiplied by itself will equal Pi.

√π = q × q = q²

√π ≈ 1.772453851

How to calculate the square root of Pi with a computer
If you are using a computer that has Excel or Numbers, then you can enter SQRT(PI) in a cell to get the square root of Pi.
Below is the result we got with 14 decimals. We call this the square root of Pi in decimal form.

SQRT(PI) ≈ 1.77245385090552

What is the square root of Pi rounded?
The square root of Pi rounded to the nearest tenth, means that you want one digit after the decimal point. The square root of Pi rounded to the nearest hundredth, means that you
want two digits after the decimal point. The square root of Pi rounded to the nearest thousandth, means that you want three digits after the decimal point.

10th: √π ≈ 1.8

100th: √π ≈ 1.77

1000th: √π ≈ 1.772

What is the square root of Pi as a fraction?
Like we said above, since the square root of Pi is an irrational number, we cannot make it into an exact fraction.
However, we can make it into an approximate fraction using the square root of Pi rounded to the nearest hundredth.

√
≈ 1.77/1
≈ 177/100
≈ 1 77/100

What is the square root of Pi written with an exponent?
All square roots can be converted to a number (base) with a fractional exponent. The square root of Pi is no exception. Here is the rule and the answer
to «the square root of Pi converted to a base with an exponent»:

√ = b^½

√ = π^½

How to find the square root of Pi by long division method
Here we will show you how to calculate the square root of Pi using the long division method.
We will calculate the square root of Pi using the first five digits of Pi which is 3.1415.

Set up 3.1415 in pairs of two digits from right to left like this:

Starting with the first set: the largest perfect square less than or equal to 3 is 1, and the square root of 1 is 1. Therefore, put 1 on top and 1 at the bottom like this:

Calculate 3 minus 1 and put the difference below. Then move down the next set of numbers.

Double the number in green on top: 1 × 2 = 2. Then, use 2 and the bottom number to make this problem:

2 × ≤ 214

The question marks are «blank» and the same «blank». With trial and error, we found the largest number «blank» can be is 7.
Replace the question marks in the problem with 7 to get:

27 × 7 = 189.

Now, enter 7 on top, and 189 at the bottom:

Calculate 214 minus 189 and put the difference below. Then move down the next set of numbers.

Double the number in green on top: 17 × 2 = 34. Then, use 34 and the bottom number to make this problem:

34 × ≤ 2515

The question marks are «blank» and the same «blank». With trial and error, we found the largest number «blank» can be is 7.
Now, enter 7 on top:

That’s it! The answer is on top. The square root of Pi with two digit decimal accuracy is 1.77. Did you notice that the last two steps repeat the previous two steps. You can add decimals by
simply adding more sets of of numbers from Pi and repeating the last two steps over and over to get a more accurate answer.

Square Root of a Number
Please enter another number in the box below to get the square root of the number and other detailed information like you got for Pi on this page.

About the online calculator

A free online calculator for basic arithmetic. This free calculator
is available for desktop and mobile devices.
Big keys and easy to use. This simple online calculator is great for anyone who doesn’t need more
advanced features.

Browser extensions for Chrome
and
Microsoft
Edge add an icon to the browser’s toolbar.
The calculator remembers your last calculation so you can close it at any time and reopen without losing
your work.

Features

• In addition to the browser extension the online calculator is available as an embeddable
  javascript widget that you can use on your web page for free
• Use this page on your phone/tablet and it will work like a mobile app. Save the page to your Home
  Screen and it will behave like a app.
• Remembers your previous calculation when re-opened
• History of calculations lets you use any of your last 10 calculations
• The desktop version lets you use the keyboard and number pad
• Copy results from the display to the clipboard

Video about using the online calculator (with tips on calculating percentages!)

Basic keys (and keyboard shortcuts)

• Number Pad or 0-9 for Number Entry
• Equals (= or Enter)
• Copy to clipboard (Ctrl-c)
• Clear («c»)
• DEL — the delete (or backspace) key removes last character you typed
• Divide (/) divide a number
• Multiply (*) multiply a number
• Minus (-) subtract a number
• Plus (+) add a number
• Plus/minus (+/-) (F3)
• Decimal point (.)
• Square x²
• Fraction ⅟x
• Parentheses for calculation order

How to calculate percentages

The % Percent (F1) key is super useful if you want to work out discounts on items in the store, and
calculating the tax on those items.

For example, you might be shopping, and want to buy something that costs $20.
but, there’s 12% tax that’s not shown in the price. By keying in 12%20 you’ll see the tax is 2.4 or
$2.40.
You can do the same calculation, by keying in 20×12%.

You can easily use the calculator to work out the total cost.
If you want to find your total cost, simply use the addition key.
key in 20+12%, you’ll get, 22.4, or $22.40.

Now, what about discounts on items?

Let’s say everything in the store is 20% off, and you want to buy something priced at $20.
Key in 20-20%, that comes to $16. But there is still 12% tax on that, so we need to add that. Key in
+12%.
So now, we’re paying $17.92.

How to calculate the square of a number

Square x² — the square key takes whatever value I key in, and squares it. That is, it will multiply it
by itself.
If I key in 9, press the square key, and then equals, we’ll get 81. That is, 9 multiplied by 9 equals
81.

Example, 9² = 81

How to calculate the square root of a number

Square Root (F2) — Square roots are the opposite of squares.

Squares are numbers getting multiplied by themselves.
The square root of a number is the value, which when multiplied by itself, gives the original number.
Squares and square roots are vice-versa methods.
For example, the square of 2 is 4 and the square root of 4 is 2.

Example, 9√ = 3, or 3+9√ = 9

How the fraction (or reciprocal) key works

Fraction ⅟x — the fraction key, or 1 over x key. (also called the reciprocal key)
If you key in 4, and then press the fraction key you will get a fraction with 1 over 4.
1 divided by 4, equals 0.25.
Example, ⅟4 = 0.25

How to use the +/- (plus/minus) key

The plus minus +/- key takes the value and converts it from a positive value to a negative
value, or a negative to a positive value.

How to use parentheses in calculations

Parentheses to control order of calculations.

Something a little more advanced, is the use of parentheses to control the order of calculations.
The two parentheses keys let you decide which part of the calculation happens first.
For example, if I key 2+3×2, we’ll get 8.
The reason it’s’ 8, is because the multiplication happens before the addition.
What I really wanted to do was add 2 to 3, and then multiply by 2.
We can do that with parentheses. Key in (3+2)*2 now the answer is 10, instead of 8.

How to use decimals

Sometimes we don’t have whole numbers and we have to deal with decimal places. Use the Decimal point (.) key for calculating decimals.
For example: 10.5+ 6 = 16.5

Quick tip: To convert a percentage to a decimal, divide by 100. Example: 35% is 35/100 or 0.35

Use this calculator to easily calculate the nth root of a given number.

What is a root of a number?

The nth-root of a number answers the question «what number can I multiply by itself n times to get this number?». It is the reverse of the exponentiation operation where the exponent is n, so if rⁿ = x, then we say that «r is the nth root of x». The mathematical operation of finding the root of a number has a special notation: the radical symbol √.

If n is even, then there are always two roots: a positive and a negative one, with equal value and opposite signs. The positive solution is called the principal root. If n is odd, then there is just one real number root and it has the same sign as x. That is its principal root. Some roots, e.g. the cube root, also have solutions in complex numbers and conjugates, but it is always the principal root which our root calculator will output.

The most popular root functions are the square root (n = 2) and cube root (n = 3), with the first one having an array of applications in mathematics, geometry, physics, probability theory and statistics. Cube roots have applications in angular calculations.

Square and cube root functions

Here are visualizations of the square root and cube root functions for a small set of integers:

The graphs were generated using this nth root calculator. It supports any root you may be interested in computing, to the extent that modern software allow.

Does the calculator support fractions?

Yes, simply enter the fraction as a decimal number (use dot as a decimal separator) and you will get the corresponding root. For example, to compute the square root of 1/2 simply enter 0.5 in the number field and 2 in the root field and you will get 0.7071 as ouput. If you are having trouble converting a fraction to a decimal number, you will find our fraction to decimal converter handy.

This free online Square Root Calculator will provide you the square root number you entered.

Math Square Root Number

In mathematics, the square root of a non-negative number a is a number that, when multiplied by itself, gives the original number a. The symbol used to represent the square root operation is √. For example, the square root of 9 is √9 = 3, because 3 multiplied by itself is equal to 9.

The square root of a number can be calculated using various methods, such as long division, prime factorization, or estimation. However, the most common method is to use a calculator or a computer program, which can perform the calculation quickly and accurately.

The square root of a non-negative number is always a non-negative number. The square root of a negative number is an imaginary number, which cannot be represented on the real number line.

The square root symbol is √ and is written before the number. For example, √9 is the square root of 9, and its value is 3.

In algebra, the square root is represented by the exponent 1/2. For example, the square root of x can be written as x^(1/2).

The square root has various applications in mathematics, science, and engineering. For example, it is used to calculate the length of the sides of a square or rectangle when the area is known. It is also used in physics to calculate the speed of a moving object or the magnitude of a force.

In addition to the square root, there are also higher order roots, such as the cube root and fourth root, which are used to find the value that, when multiplied by itself multiple times, gives the original number.

Math Square Number

Square numbers also have practical applications in geometry and algebra, where they are used to calculate areas of squares and solve equations involving quadratic functions.

How to use the Online Math Square Root Calculator?

1. Browse or open Online Math Square Root Calculator — https://www.onlinewebtoolkit.com/square-root-calculator
2. Enter your number, on the left side, in place of the √X
3. Click the «Calculate Square Root» button.
4. This tool will calculate and display result in other textbox on the fly.

Useful Features of Our Online Math Square Root Calculator

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