Square root of 32 : Simplified Radical Form

To find the value of the square root of 32, we need to first understand the concept of square roots. We know that the square root of a number $x$ is denoted by $\sqrt{x}.$ So √32 is the square root of 32, and by definition, it is a number $x$ when multiplied by itself will be 32. Thus we need to find the number $x$ such that \[x \times x=32.\]

Overview of Square Root of 32

  • The value of square root of 32 is 5.65686…
  • Square root of 32 in radical form: √32
  • Note that √32 is a quadratic surd.
  • 321/2 is the exponential form of square root of 32
  • Square root of 32 is irrational.
  • √32=4√2
  • Square root of 32 is 5.6569 up to 4 decimal.
  • Square root of 32 is not a whole number.
  • 32 is not a perfect square
  • 32 square: 322 =32 × 32 =1024

Simplify Square Root of 32

What is the simplified radical form of square root of $32$?

Note that 32 = 16×2.

So $\sqrt{32}$ $=\sqrt{16 \times 2}$

$=\sqrt{16} \times \sqrt{2}$ $[\because \sqrt{x \times y}=\sqrt{x} \times \sqrt{y}]$

As $\sqrt{16}=4,$ we obtain that

$\sqrt{32}$$=4 \times \sqrt{2}$ $=4\sqrt{2}.$ 

4√2 is the simplified form of square root of 32.

 

Value of Square Root of 32

Simplifying the surd $\sqrt{32},$ we will get the value of square root of $32.$ From above we get that $\sqrt{32}$ $=4 \times \sqrt{2}.$  We know that $\sqrt{2}=1.414.$ Putting this value, we have

$\sqrt{32}=4 \times 1.414$

$=5.656$

So the value of square root of $32$ is $=5.656.$

 

Is Square Root of 32 Rational?

We know that the square root of 32 is 4√2, that is, √32=4√2.

As √2 is an irrational number, we conclude that 4√2 is not a rational number.

∴ The square root of 32 is also an irrational number.

So √32 is not a rational number.

 

Square Root of 32 by Prime Factorization

At first, we will try to find the prime factorization of $32.$ As $32$ is an even number, $2$ will divide it. So we have

$32=2 \times 16.$

In a similar way, $16=2 \times 8.$

As $8=2 \times 4$ and $4=2 \times 2,$ we finally get that

$32=2 \times 2 \times 2 \times 2 \times 2$ $\cdots (\star)$ 

As $2$ is a prime number, the above is the prime factorization of $32.$ Now we will take square root on both sides of $(\star)$, and doing that we get

32=(2×2×2×2×2)

=(2×2) ×(2×2) ×2 as we know that √(x×y×z)=√x ×y ×z. Here x=2×2, y=2×2 and z=2

= 2 × 2 ×2

= 42

Thus, the square root of 32 is 42 obtained by the prime factorization method.

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