Square root of 64

The square root of 64 is a number when multiplied by itself will be 64. It is the positive solution of the quadratic equation x²=64. The square root of 64 is denoted by the symbol √64. In this section, we will learn how to find the value of √64.

Note that if a square has an area of 64 unit², then we use the square root of 64 to find the length of the square.

The value of the square root of 64 is 8.

If you want to know more about square roots, visit the following pages:

• Basic concepts of square roots
• Properties of square roots

Important Things:

• √64 = 8
• 64 is a perfect square.
• √64 is the radical form of square root of 64.
• 64^1/2 is the exponential form of square root of 64.
• The square root of 64 is 8.000 in decimal form.
• Square of 64: 64×64 = 4096

It’s time to evaluate the square root of 64.

What is the Square Root of 64?

Note that the value of the square root of 64 is 8. To prove this, we will write 64 as a square of some numbers as 64 is a perfect square.

See that 64 is a product of two numbers of 8‘s. So we write

64=8×8=8²

Taking square root on both sides, we have

√64 = √8²

As square root can be written as power 1/2, we get that

√64 = (8²)^1/2 = 8^2×1/2   as we know that (a^m)ⁿ = a^m×n

= 8¹ = 8

So the square root of 64 is 8.  

Is 64 a perfect square number?

Note that √64=8 and 8 is a whole number. This makes the square root of 64 is also a whole number.

∴ the definition of a perfect square implies that 64 is a perfect square number.

 

Square root of 64 by Prime Factorization

Now we will use the prime factorization method to find the square root of 64.

At first, we have to factorize 64. As 64 is an even number, it is divisible by 2. So we have

64=2×32

Again as 32 is an even number, we have

32=2×16

Similarly, 16=2×8,  8=2×4  and  4=2×2.

So finally we get 

64=2×2×2×2×2×2 …(∗)

As 2 is a prime number, the above is the prime factorization of 64.

Taking square root on both sides of (∗), we get that

√64 = √(2×2×2×2×2×2)

= √(2×2) × √(2×2) × √(2×2)

= 2×2×2  as we know that √a×a=a

= 8 

∴  the square root of 64 is 8. 

Is Square Root of 64 Rational?

First, recall the definition of a rational number. A number of the form p/q where p and q are integers, q ≠ 0, is called a rational number.

We have shown above that √64=8.

Note that 8=8/1.

∴ 8 can be written as p/q with p=8, q=1≠0.

This makes 8 a rational number.

So the square root of 64 is also a rational number.

Conclusion: √64 is not an irrational number.

 

Key Things about 64

• 64 is a composite odd number.
• A square of length 8 unit has the area 64 unit²