506 LearnersLast updated on May 26th, 2025
Square Root of 64
The square root of 64 is a value “y” such that when “y” is multiplied by itself → y ⤫ y, the result is 64. The number 64 has a unique non-negative square root, called the principal square root.
What is the Square Root of 64?
The square root of 64 is ±8, where 8 is the positive solution of the equation x2 = 64. Finding the square root is just the inverse of squaring a number and hence, squaring 8 will result in 64.
The square root of 64 is written as √64 in radical form, where the ‘√’ sign is called the “radical” sign. In exponential form, it is written as (64)1/2
Finding the Square Root of 64
We can find the square root of 64 through various methods. They are:
- Prime factorization method
- Long division method
- Repeated subtraction method
Square Root of 64 By Prime Factorization Method
The prime factorization of 64 can be found by dividing the number by prime numbers and continuing to divide the quotients until they can’t be separated anymore, i.e., we first prime factorize 64 and then make pairs of two to get the square root.
So, Prime factorization of 64 = 8 × 8
Square root of 64 = √[8 × 8] = 8
Square Root of 64 By Long Division Method
This method is used for obtaining the square root for non-perfect squares, mainly. It usually involves the division of the dividend by the divisor, getting a quotient and a remainder too sometimes.
Follow the steps to calculate the square root of 64:
Step 1: Write the number 64 and draw a bar above the pair of digits from right to left.
64 is a 2-digit number, so it is already a pair.
Step 2: Now, find the greatest number whose square is less than or equal to 64. Here, it is 8
Because 82=64
Step 3: Now divide 64 by 8 (the number we got from Step 2) and we get a remainder of 0.
Step 4: The quotient obtained is the square root. In this case, it is 8.
Square Root of 64 By Subtraction Method
We know that the sum of the first n odd numbers is n2. We will use this fact to find square roots through the repeated subtraction method. Furthermore, we just have to subtract consecutive odd numbers from the given number, starting from 1. The square root of the given number will be the count of the number of steps required to obtain 0. Here are the steps:
Step 1: take the number 64 and then subtract the first odd number from it. Here, in this case, it is 64-1=63
Step 2: we have to subtract the next odd number from the obtained number until it comes zero as a result. Now take the obtained number (from Step 1), i.e., 63, and againsubtract the next odd number after 1, which is 3, → 63-3=60. Like this, we have to proceed further.
Step 3: now we have to count the number of subtraction steps it takes to yield 0 finally.
Here, in this case, it takes 8 steps
So, the square root is equal to the count, i.e., the square root of 64 is ±8.
Common Mistakes and How to Avoid Them in the Square Root of 64
When we find the square root of 64, we often make some key mistakes, especially when we solve problems related to that. So, let’s see some common mistakes and their solutions.
Mistake 1
Misunderstanding symbol.
Often, when √64 is mistaken as 642, we square the number and get the result as 4096. So, understanding of symbol should be clear.
Mistake 2
Ignoring the negative root
Remember that both +8 and -8 are valid square roots of 64 where the principal square root is 8, but -8 is also correct.
Mistake 3
Misinterpreting the square root symbol
√64 refers to the positive principal square root. ±√64 will yield both +8 and -8.
Mistake 4
Applying square root to non-perfect squares
You have to confirm that 64 is a perfect square before expecting the root will be an integer
Mistake 5
Forgetting to check for perfect squares
Check if 64 is a perfect square. Since, 64 = 82., it is confirmed that 64 is a perfect square
Square Root of 64 Examples
Problem 1
Find the radius of a circle whose area is 64π² cm²
Given, the area of the circle = 64π cm2
Now, area = πr2 (r is the radius of the circle)
So, πr2 = 64π cm2
\We get, r2 = 64 cm2
r = √64 cm
Putting the value of √64 in the above equation,
We get, r = ±8 cm
Here we will consider the positive value of 8.
Therefore, the radius of the circle is 8 cm.
Answer: 8 cm
Explanation
We know that, area of a circle = πr2 (r is the radius of the circle) According to this equation, we are getting the value of “r” as 8 cm by finding the value of the square root of 64.
Problem 2
Find the length of a side of a square whose area is 64 cm²
Given, the area = 64 cm2
We know that, (side of a square)2 = area of square
Or, (side of a square)2 = 64
Or, (side of a square)= √64
Or, the side of a square = ± 8.
But, the length of a square is a positive quantity only, so, the length of the side is 8 cm.
Answer: 8 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 its square root is the measure of the side of the square.
Problem 3
Simplify the expression: √64 ╳ √64, √64+√64
√64 ╳ √64
= √(8 ╳ 8) ╳ √(8 ╳ 8)
= 8 ╳ 8
= 64
√64+√64
= √(8 ╳ 8) + √(8 ╳ 8)
= 8 + 8
= 16
Answer: 64, 16
Explanation
In the first expression, we multiplied the value of the square root of 64 with itself. In the second expression, we added the value of the square root of 64 with itself.
Problem 4
If y=√64, find y²
firstly, y=√64= 8
y2=82=64
or, y2=64
Answer : 64
Explanation
Squaring “y” which is same as squaring the value of √64 resulted to 64.
Problem 5
Calculate (√64/4 + √64/2)
√64/4 + √64/2
= 8/4 + 8/2
= 2 + 4
=6
Answer : 6
Explanation
From the given expression, we first found the value of square root of 64 then solved by simple divisions and then simple addition. We conclude that the square root of 64 is derived by multiplying 8 with itself, i.e., 8 ╳ 8. The relation between square and square root is that they are inverse of each other.
FAQs on 64 Square Root
1.Can the square root of 64 be negative?
2.What is the cube root of 64?
3.Is 64 a perfect cube?
4.Is 64 a perfect square or a non-perfect square?
5.Is the square root of 64 a rational or irrational number?
6.What is the principal square root of 64?
7.How to solve √63?
Important Glossaries for Square Root of 64
- 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.Ex: 2 ⤬ 2 ⤬ 2 ⤬ 2 = 16 Or, 2 4 = 16, where 2 is the base, 4 is the exponent
- Factorization : Expressing the given expression as a product of its factors Ex: 48=2 ⤬ 2 ⤬ 2 ⤬ 2 ⤬ 3
- Prime Numbers :Numbers which are greater than 1, having only 2 factors as →1 and Itself. Ex: 1,3,5,7,....
- Rational numbers and Irrational numbers:The Number which can be expressed as p/q, where p and q are integers and q not equal to 0 are called Rational numbers. Numbers which cannot be expressed as p/q, where p and q are integers and q 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. Ex: 4,9,25 Non-perfect square numbers are those numbers whose square roots comprise decimal places. Ex :3, 8, 24.
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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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