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Express each one of the following with rational denominator:$ \frac{3 \sqrt{2}+1}{2 \sqrt{5}-3} $
Given:
\( \frac{3 \sqrt{2}+1}{2 \sqrt{5}-3} \)
To do:
We have to express the given fraction with rational denominator.
Solution:
We know that,
Rationalising factor of a fraction with denominator ${\sqrt{a}}$ is ${\sqrt{a}}$.
Rationalising factor of a fraction with denominator ${\sqrt{a}-\sqrt{b}}$ is ${\sqrt{a}+\sqrt{b}}$.
Rationalising factor of a fraction with denominator ${\sqrt{a}+\sqrt{b}}$ is ${\sqrt{a}-\sqrt{b}}$.
Therefore,
$\frac{3 \sqrt{2}+1}{2 \sqrt{5}-3}=\frac{(3 \sqrt{2}+1)(2 \sqrt{5}+3)}{(2 \sqrt{5}-3)(2 \sqrt{5}+3)}$
$=\frac{3 \sqrt{2} \times 2 \sqrt{5}+3 \times 3 \sqrt{2}+2 \sqrt{5}+3}{(2 \sqrt{5})^{2}-(3)^{2}}$
$=\frac{6 \sqrt{10}+9 \sqrt{2}+2 \sqrt{5}+3}{20-9}$
$=\frac{6 \sqrt{10}+9 \sqrt{2}+2 \sqrt{5}+3}{11}$
Hence, $\frac{3 \sqrt{2}+1}{2 \sqrt{5}-3}=\frac{6 \sqrt{10}+9 \sqrt{2}+2 \sqrt{5}+3}{11}$.