Question

How do you simplify the square root of $240$?

Answer 1

Hint:
Square root of a number can be calculated by prime factorization, the number where factors are the numbers which completely divide the given number with no remainder. Prime factorization is defined as expressing a number as a product of prime numbers.

Complete step by step solution:
Given the number to simplify is $\sqrt{240}$.
First to get the square root of $240$ , we will try to find its prime factors.
The prime factor of $240$ will be ,
 $2|240$
 $2|120$
 $2|60$
 $2|30$
 $3|15$
 $5|5$
 $1$
Therefore, $240$ has 2 , 3 and 5 as its prime factors.
Hence the factor of $240=2\times 2\times 2\times 2\times 3\times 5$ .
Now we will make pairs of similar factors as they can be taken out of the square root.
Therefore,
$\sqrt{240}=\sqrt{2\times 2\times 2\times 2\times 3\times 5}$
$\sqrt{240}=\sqrt{\overline{2\times 2}\times \overline{2\times 2}\times 3\times 5}$
After forming a pair of the similar factors, we will take a pair out of the square root and thus, continue the process to simplify and attain the answer.
$\sqrt{240}=2\times 2\sqrt{3\times 5}$
$\sqrt{240}=4\sqrt{15}$

Hence, we get the square root of $240$ as $4\sqrt{15}$.

Note:
Square root of any number is calculated by prime factorization of the number. The student should properly know the concepts of prime factorization to solve these kinds of questions. Another way to simplify the square root of any number is using the long division.