Square Root of 8
2026-02-28 01:40 Diff
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Last updated on October 30, 2025

The square root of 8 is a value “y” such that when “y” is multiplied by itself → y ⤫ y, the result is 8. The number 8 has a unique non-negative square root, known as the principal square root.

What Is the Square Root of 8?

The square root of 8 is ±2.82842712475, where 2.82842712475 is the positive solution of the equation x2 = 8. Finding the square root is just the inverse way of squaring a number, and hence, squaring 2.82842712475 will result in 8.

The square root of 8 is written as √8 in radical form, where the ‘√’  sign is called the “radical”  sign. In exponential form, it is written as (8)1/2.

Square Root of 8 By Prime Factorization

The prime factorization of 8 can be found by dividing the 8 by prime numbers and continuing to divide the quotients until they can’t be divided anymore. After factorizing 8, make pairs out of the factors to get the square root. If there exist numbers that cannot be made pairs further, we place those numbers with a “radical” sign along with the acquired pairs.

So, Prime factorization of 8 = 2 × 2 × 2 = 2³

But here, only a pair of factor 2 can be obtained and a single 2 is remaining

So, it can be written as  √8 =  2√2.

Square root of 8 = √[2 × 2 × 2] = 2√2, i.e., 2.82842712475

Square Root of 8 By Long Division

This is a method, mainly used for obtaining the square root for non-perfect squares. It usually involves the division of the dividend by the divisor, getting a quotient and a remainder too, where the dividend is the number we are finding the square root of.

Follow the steps to calculate the square root of 8:

 Step 1: Place the number 8, just the same as the image, starting from right to left, and draw a bar above the pair of digits since 8 is a 1-digit number, so simply just draw a bar above 8. 

Step 2: Now, find the greatest number whose square is less than or equal to 8. Here, it is 2, because 22=4 < 8

Step 3: Now divide 8 by 2 (the number we got from step 2) and we get a remainder. Double the divisor 2, we get 4 and find the largest possible number A, put it in the right of 4, through this, a double-digit is formed. Now multiply this with the same number A. Repeat this process until you reach the remainder of 0.

Step 4: The quotient we got is the square root. In this case, it is 2.828….

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Square Root of 8 By Approximation

Follow the steps below: 

Step 1: find the square roots of the perfect squares above and below 8

Below : 4 → square root of 4 =2  …. (i)

Above : 9 →square root of 9 = 3 …..(ii)

Step 2: Dividing 8 with one of 2 or 3 

If we choose 3 and divide 8 by 3, we are getting 2.666 …..(iii)

Step 3:  find the average of 3 (from step (ii)) and 2.666 (from step (iii))

(3+2.666)/2 = 2.8333 

Hence, 2.8333 is the approximate square root of 8

Similarly, try applin these methods on square root of 100 and practice.

Common Mistakes and How to Avoid Them in the Square Root of 8

When we find the square root of 8, we often make some key mistakes, especially when we solve problems related to that. So, let’s see some common mistakes and their solutions.

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Problem 1

Estimate the value of √8 using an initial guess of 2.5

Okay, lets begin

By using the formula,


New Guess=(Initial Guess + (Given Number / Initial Guess))/ 2


applying the formula, 


New guess= (2.5 + (8/2.5))/2 


                 = (2.5+ 3.2)/2


                 =5.7/2


                 =2.85


Again, repeating the process,


New guess= (2.85 + (8/2.85))/2 


                 = (2.85+ 2.807)/2


                 =5.657/2


                 =2.825


hence, after a few iterations, the value of √8 is approximately 2.825

Answer: 2.825 approx.
 

Explanation

Using the formula for New Guess, we found the approximate value of the square root 8 by repeated iterations, where  New Guess=(Initial Guess + (Given Number / Initial Guess))/ 2
 

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Problem 2

Find the length of a side of a square whose area is 8 cm².

Okay, lets begin

 Given, the area = 8 cm2


We know that, (side of a square)2 = area of square


 Or,  (side of a square)2 = 8


Or,  (side of a square)= √8


Or, side of a square = ± 2√2.


But, the length of a square is a positive quantity only, so, the length of the side is 2√2 cm.

Answer: 2√2 cm

Explanation

We know that, (side of a square)2 = area of square. Here, we are given with the area of the square, so, we can easily find out its square root because the square root is the measure of the side of the square.

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Problem 3

Simplify (√8 + √8) ⤫ √8

Okay, lets begin

 (√8 + √8) ⤫ √8 = (2√2 + 2√2) ⤫ 2√2 = 4√2  ⤫ 2√2 = 8 ⤫ 2 =16 

Answer: 16

Explanation

We first solved the part inside the brackets, i.e., √8 + √8, which resulted into 4√2, and then multiplying it with √8 which is 2√2, we get 16.

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Problem 4

If x= √8, what is x² -3?

Okay, lets begin

x= √8

⇒ x2 = 8

⇒ x2-3 = 8-3

⇒ x2-3 = 5

Answer: 5

Explanation

We did the square of the given value of x and then subtracted 3 from it.

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Problem 5

If y=√8, find y²

Okay, lets begin

Firstly, y=√8= 2.8284,

Now, squaring y, we get, y2= (2.8284)2=8, or, y2=8

Answer: 8

Explanation

Squaring “y” which is the same as squaring the value of √8 resulted as 8.

Well explained 👍

FAQs on Square Root 8

1.Can the value of √8 be negative?

Yes, the square root of 8 can be negative. Rather, it can be both positive and negative. So, (-2√2)² = 8 and (2√2)²= 8, both yield the same result.

2.Is the square root of 8 a whole number?

3.Is 8 a perfect square or a non-perfect square?

8 is a non-perfect square, since 8 =(2√2) ².

4.Is the square root of 8 a rational or irrational number?

The square root of 8 is ±2√2. So, 2√2 is an irrational number since it cannot be obtained by dividing two integers and cannot be written in the form p/q.

5.What is the principal square root of 8?

The principal square root of 8 is ±2√2, which is the positive value, whereas  -2√2 is not.

Important Glossaries for Square Root of 8

  • Exponential form: An algebraic expression that includes an exponent. It is a way of expressing the numbers raised to some power of their factors. It includes continuous multiplication involving base and exponent.
  • Factorization: Expressing the given expression as a product of its factors
  • Prime Numbers: Numbers that are greater than 1, having only 2 factors as →1 and Itself. 
  •   Rational numbers and Irrational numbers: The numbers that can be expressed as p/q, where p and q are integers and q is not equal to 0 are called Rational numbers. Numbers that cannot be expressed as p/q, where p and q are integers and q is not equal to 0 are called Irrational numbers. 
  • Perfect and non-perfect square numbers: Perfect square numbers are those numbers whose square roots do not include decimal places.  Non-perfect square numbers are those numbers whose square roots comprise decimal places. 

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Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

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