Question

How do you simplify $\sqrt{\dfrac{98}{{{x}^{2}}}}$?

Answer 1

Hint: In this question we have been given a term which is in the square root. The term consists of a fraction. We will use the various properties for terms under square root and simplify the given term. We will split the term in the numerator such that it is the multiple of a real square number and then use the property $\sqrt{{{a}^{2}}}=a$, to remove it out of the square root.

Complete step-by-step solution:
We have the expression given to us as:
$\Rightarrow \sqrt{\dfrac{98}{{{x}^{2}}}}$
Now we know the property that $\sqrt{\dfrac{a}{b}}=\dfrac{\sqrt{a}}{\sqrt{b}}$ therefore, on using this property, we get:
$\Rightarrow \dfrac{\sqrt{98}}{\sqrt{{{x}^{2}}}}$
Now we can see in the denominator that the term is in the form $\sqrt{{{a}^{2}}}$ therefore, on using the property $\sqrt{{{a}^{2}}}=a$ and taking it out of the square root, we get:
$\Rightarrow \dfrac{\sqrt{98}}{x}$
Now we can write the term $98=49\times 2$ therefore, on substituting, we get:
$\Rightarrow \dfrac{\sqrt{49\times 2}}{x}$
Now we know that the term $49$ has a real square which is $7$, therefore we can write the expression as:
$\Rightarrow \dfrac{\sqrt{{{7}^{2}}\times 2}}{x}$
On using the property $\sqrt{{{a}^{2}}}=a$ in the numerator, we get:
$\Rightarrow \dfrac{7\sqrt{2}}{x}$, which is the required solution.

Note: It is to be remembered that the $\sqrt{{}}$ symbol is called as the radical symbol and it is used to represent a radical expression. In this question the radical expression was in the form of the square root of the term, it can also be in the form of cube root, fourth root etc. the terms in square root can also be written using the exponential form as $\sqrt{a}={{a}^{1/2}}$.