Question

How do you simplify the square root of 72?

Answer 1

Hint: To simplify the square root of 72, we have to prime factorize 72. We will have to pair the same prime factors. The unpaired ones will be multiplied inside the square root while the paired ones will be taken outside as a single number and will be multiplied outside.

Complete step by step solution:
We have to simplify the square root of 72. Let us first prime factorize 72 using the division method. For this, we will write 72 to the right and the prime factor that completely divides 72 to the left as shown below. We have to write the quotient below 72.
$\begin{array}{rl}& 2|\underset{―}{72}\\ & 2|\underset{―}{36}\\ & 2|\underset{―}{18}\\ & 3|\underset{―}{9}\\ & 3|\underset{―}{3}\\ & 1\end{array}$
Let us write the prime factors as follows.
$\Rightarrow 72=2\times 2\times 2\times 3\times 3$
We have to pair the same prime factors. The unpaired ones will be multiplied inside the square root while the paired ones will be taken outside as a single number and will be multiplied outside.
$\Rightarrow 72=\boxed{{\displaystyle 2\times 2}}\times 2\times \boxed{{\displaystyle 3\times 3}}$
Hence, we can write the $\sqrt{72}=2\times 3\times \sqrt{2}=6\sqrt{2}$ .

Hence, the simplified form of square root of 72 is $6\sqrt{2}$ .

Note: Students must know how to prime factorize numbers. We can see that 72 is not a perfect square because a perfect square will have an even number of factors. We can also find the square root of 72 in an alternate method. Let us write 72 as a product of two numbers where each or one of these numbers are perfect squares. We can write 72 as a product of 2 and 36.
$\Rightarrow \sqrt{72}=\sqrt{2\times 36}$
We know that $\sqrt{a\times b}=\sqrt{a}\times \sqrt{b}$ . Hence, we can write
$\Rightarrow \sqrt{72}=\sqrt{2}\times \sqrt{36}$
We know that 36 is a perfect square, that is, $6^2=36$ . On taking square root, we will get
$\Rightarrow \sqrt{72}=\sqrt{2}\times 6=6\sqrt{2}$