Question

How do you rationalise the denominator and simplify \[\dfrac{3\sqrt{9}}{\sqrt{27}}\] ?

Answer 1

Hint: In this question, we need to rationalise the denominator and simplify the given expression \[\dfrac{3\sqrt{9}}{\sqrt{27}}\] . Rationalising the denominator is nothing but getting rid of any square roots or any cubic roots by multiplying with its conjugate. First, we can separately simplify the numerator and the denominator of the given expression. Then we can rationalise the simplified expression by multiplying with the conjugate of the denominator.

Complete step by step answer:
Given, \[\dfrac{3\sqrt{9}}{\sqrt{27}}\]
Now we can simplify the expression .
First we can simplify the numerator term,
\[\sqrt{9} = \sqrt{3 \times 3}\]
On taking the terms out of radical sign,
We get,
\[\sqrt{9} = 3\]
Then we can simplify the denominator term,
\[\sqrt{27} = \sqrt{3 \times 3 \times 3}\]
On taking the terms out of radical sign,
We get,
\[\sqrt{27} = 3\sqrt{3}\]
Thus the expression becomes,
\[\dfrac{3\sqrt{9}}{\sqrt{27}} = \dfrac{3 \times 3}{3\sqrt{3}}\]
On simplifying,
We get,
\[\Rightarrow \dfrac{3}{\sqrt{3}}\]
Now we can rational the denominator,
The conjugate of \[\sqrt{3}\] is \[\sqrt{3}\] .
In order to rationalise the expression, multiply and divide the expression by \[\sqrt{3}\] .
\[\Rightarrow \dfrac{3}{\sqrt{3}} = \dfrac{3 \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}}\]
On multiplying,
We get,
\[\Rightarrow \ \dfrac{3\sqrt{3}}{3}\]
On simplifying,
We get,
\[\Rightarrow \ \sqrt{3}\]
Thus the value of \[\dfrac{3\sqrt{9}}{\sqrt{27}}\] is \[\sqrt{3}\]

Note: A conjugate is nothing but a similar term but with a different and opposite sign. The conjugate of \[\left( 2 + \ \sqrt{8} \right)\] is \[\left( 2 - \ \sqrt{8} \right)\] . In the process of rationalizing the denominator of the given expression , the conjugate is the rationalizing factor and the aim of students should be to convert the denominator in simplified form. Generally in denominator we try to form an expression similar to the identity (a+b)(a-b).