Question

Rationalize the denominator of the following $\dfrac{1}{\sqrt{7}}$.

Answer 1

Hint: In the above-given expression, the rationalization of a denominator is done by multiplying and dividing the whole expression by the conjugate of $\sqrt{7}$ which is itself $\sqrt{7}$. Then do the multiplication to get the desired result.

Complete step by step answer:
To rationalize the denominator, we will multiply and divide the whole expression by the conjugate of the denominator is $\sqrt{7}$ itself as there is no rational part added or subtracted. On multiplying the numerator and denominator by $\sqrt{7}$, we will get:
$\Rightarrow \dfrac{1}{\sqrt{7}}\times \dfrac{\sqrt{7}}{\sqrt{7}}$
Multiply the terms in the numerator and denominator,
$\Rightarrow \dfrac{\sqrt{7}}{{{\left( \sqrt{7} \right)}^{2}}}$
Square the term in the denominator,
$\Rightarrow \dfrac{\sqrt{7}}{7}$

Hence, the required answer is $\dfrac{\sqrt{7}}{7}$.

Note:
You might be wondering as it is a question's requirement so we have rationalized this expression. But in general, why is there a need to rationalize? The answer is as you can see that after rationalization meaning multiplying and dividing the whole expression by a conjugate, the denominator of the expression is reduced to rational by using basic algebraic identities. So, in a calculation, if you find the denominator can be rationalized then go for it, as it will reduce the complexity of the problem.
Rationalization is the process of eliminating a radical or imaginary number from the denominator of an algebraic fraction. That is, remove the radicals in a fraction so that the denominator only contains a rational number. Suppose the denominator contains a radical expression $a+\sqrt{b}$ or $a+i\sqrt{b}$, the fraction must be multiplied a quotient containing $a-\sqrt{b}$ or $a-i\sqrt{b}$.