Question

What are the steps for simplifying radicals?

Answer 1

Hint: To write the steps for simplifying radicals we need to know what radicals are. Radicals are something which is written in either $\sqrt{{}}$ or ${{\left( ^{{}} \right)}^{\dfrac{1}{n}}}$. Suppose we have given a radical form as: $\sqrt{40}$. So, to simplify that we are going to first of all prime factorize what is written inside this radical. Then the square root is given in $\sqrt{40}$ so we are trying to club the two numbers of the same kind. Now, we will remove the pair from inside the radical to outside the radical. There are some details in these steps which we are going to explain in the below.

Complete step-by-step solution:
Radicals are expressed in the following form:
$\sqrt{{}}$ or ${{\left( ^{{}} \right)}^{\dfrac{1}{n}}}$
To write the steps for the simplification of radicals, we are going to use the following example:
$\sqrt{40}$
Now, we are going to first of all do the prime factorization of the number written in the radical (which is a square root).
The prime factorization for 40 is as follows:
$40=2\times 2\times 2\times 5$
Now, we are going to write this prime factorization inside the radical form:
$\sqrt{2\times 2\times 2\times 5}$
As the radical given is a square root so we have to pair the two numbers of the same kind. In the below, we have underlined the pair:
$\sqrt{\underline{2\times 2}\times 2\times 5}$
Now, we are going to take this underlined pair out from the square root and when we take this pair out then only one of the number will remain and it will look as follows:
$2\sqrt{2\times 5}$
After that multiply whatsoever is written inside the square root and we get,
$2\sqrt{10}$
Similarly, you can find for the radical of the following type:
$\sqrt[3]{54}$
Writing the prime factorization of 54 we get,
$\sqrt[3]{3\times 3\times 3\times 2}$
As the radical is a cube root so we have to take three numbers of the same kind so we are underlining the three numbers of the same kind as follows:
$\sqrt[3]{\underline{3\times 3\times 3}\times 2}$
Taking the underlining numbers out from the radical expression we get,
$3\sqrt[3]{2}$
In this way, we can simplify the radicals stepwise.


Note: The simplification for the radicals will be correct when we correctly write the prime factorization of the number written inside the radical and then properly take the numbers of the same kind. Failure of any such concept will lead you to the wrong answer.