Question

How do you simplify the square root of $162$?

Answer 1

Hint: In order to find the square root of the given number, first we need to solve the square root by expanding the radicand in terms of its prime factors and then taking out the common values whose square roots are known. Or, we know that in square root, for two same numbers inside the bracket only the number that comes out of the bracket.

Complete step by step solution:
We are given the number $162$.
Writing this value in the terms of square root: $\sqrt {162} $.
Solving the radicand that is $\sqrt {162} $ by expanding it in terms of its prime factors and we get that the prime factors of $162$ is $2 \times 3 \times 3 \times 3 \times 3$.
Putting the factors inside the Square root:
$\sqrt {162} = \sqrt {2 \times 3 \times 3 \times 3 \times 3} $.
Since, we know that two similar numbers present inside the root can be taken out as a single unit, what we call its square root. For ex: $\sqrt {x \times x} = x$.
Similarly, we know that $\sqrt 9 = \sqrt {3 \times 3} = 3$.
As we can see that there are $3 \times 3 \times 3 \times 3$, so we can take out two 3’s from the root and we are left with $2$inside that is: $\sqrt {2 \times 3 \times 3 \times 3 \times 3} = 3 \times 3\sqrt 2 = 9\sqrt 2 $.
Since, $2$is a single unit and it has no square root, so it cannot be taken out from the root, which means we cannot further simplify the root.
Therefore, simplest radical form or the square root of $\sqrt {162} $ is $9\sqrt 2 $.

Note:
1. We can leave the value inside the roots as it is or we can find its decimal value and multiply it with the outer value.
2. If the radicand cannot be further expanded or it’s a prime number then leave it as it is.
3. Always cross check the answer once.
4. Radicand is the term or value that is present inside the roots.