Question

How do you simplify $\sqrt {165} $?

Answer 1

Hint:
Here we must know that procedure to take the square root of any number. When we take the square root of any number let us say $a$ we denote it by the symbol $\sqrt a $ and then we actually need to find the number which when multiplied by itself gives us the number $a$. Any factor of it which is in pair can be taken out of the root and written once.

Complete step by step solution:
Here we are given to find the square root of the number which is given us as $165$
Now first of all, we need to know what the square root means. Square root of any number let us say $a$ we denote it by the symbol $\sqrt a $ and then we actually need to find the number which when multiplied by itself gives us the number$a$.
First step here is to write the number whose root is to be found in the form of multiplication of its factors.
For example: If we have the number $12$ then we can write it as $(2) \times (2) \times (3)$
Now if we had to take its root we would have written it as $\sqrt {12} = \sqrt {(2)(2)(3)} $
As the pair of one $2$ is inside the root so we can take one two outside the root and write it as $2\sqrt 3 $
Similarly we need to write the factor form of $165$
$
  165 = 3 \times 55 \\
  55 = 5 \times 11 \\
  11 = 11 \times 1 \\
 $
So we can write that there are three factors of $165$ as $3,5,11$
$\sqrt {165} = \sqrt {(3)(5)(11)} $
Now we can see that there is not any digit out of $3,5,11$ which are in pair so we cannot simplify it further and therefore the simplified form is also $\sqrt {165} $
Hence we cannot simplify it further.

Note:
Here we can also find the exact value of this root using the calculator where we just need to multiply the roots of $3,5,11$ and get the value required. If we are given to find the cube root of any number then we need to write outside the cube root that number which is occurring three times instead of two times.