Question

How do you rationalise the denominator and simplify \[\sqrt{\dfrac{5}{3}}\] ?

Answer 1

Hint: In this question, we need to rationalise the denominator and simplify the given expression \[\sqrt{\dfrac{5}{3}}\] . Rationalising the denominator is nothing but getting rid of any square roots or any cubic roots by multiplying with its conjugate. First we can rewrite the given expression by using the rule \[\sqrt{\dfrac{a}{b}} = \dfrac{\sqrt{a}}{\sqrt{b}}\] . Then we can multiple both the numerator and denominator by \[\sqrt{3}\] . Since the conjugate of \[\sqrt{3}\] is \[\sqrt{3}\] . Then we need to just simplify the expression by using basic rules of simplification.

Complete step-by-step answer:
Given, \[\sqrt{\dfrac{5}{3}}\]
We need to rationalise the denominator and simplify the given expression.
In the given expression, the denominator is \[\sqrt{3}\] . We can rationalise the denominator of the fraction by multiplying both the numerator and denominator by its conjugate.
Given, \[\sqrt{\dfrac{5}{3}}\]
We can rewrite \[\sqrt{\dfrac{5}{3}}\] as \[\dfrac{\sqrt{5}}{\sqrt{3}}\] .
\[\Rightarrow \dfrac{\sqrt{5}}{\sqrt{3}}\]
The conjugate of \[\sqrt{3}\ \] is \[\sqrt{3}\]
Now we can multiply both the numerator and denominator by \[\sqrt{3}\] .
\[\Rightarrow \dfrac{\sqrt{5}}{\sqrt{3}} \times \dfrac{\sqrt{3}}{\sqrt{3}}\]
On simplifying,
We get,
\[\Rightarrow \dfrac{\sqrt{15}}{\sqrt{9}}\]
On observing the numerator term \[\sqrt{15}\] is a perfect square number thus we can’t simplify it further. But \[\sqrt{9}\] is a perfect square number
Thus we can simplify the denominator term, as
\[\sqrt{9} = \sqrt{3 \times 3}\ \]
\[\Rightarrow \dfrac{\sqrt{15}}{\sqrt{3 \times 3}}\]
On taking the terms out of radical sign,
We get,
\[\Rightarrow \dfrac{\sqrt{15}}{3}\]
Thus we get the rationalized form of \[\sqrt{\dfrac{5}{3}}\] is \[\dfrac{\sqrt{15}}{3}\] .
Final answer :
 The rationalized form of \[\sqrt{\dfrac{5}{3}}\] is \[\dfrac{\sqrt{15}}{3}\] .

Note: We need to know that a type of numbers that are divided into two parts by a horizontal line is known as fraction. The top of the fraction is known as the numerator and the bottom of the fraction is called the denominator. Denominator consists of both rational and irrational numbers. Numbers like \[6\] and \[7\] are rational numbers and similarly numbers like \[\sqrt{6}\] and \[\sqrt{7}\] are irrational numbers. A conjugate is nothing but a similar term but with a different and opposite sign. The conjugate of \[\left( 3 + \ \sqrt{7} \right)\] is \[\left( 3 - \ \sqrt{7} \right)\] . In this process of rationalizing the denominator of the given expression , the conjugate is the rationalizing factor.