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Express each one of the following with rational denominator:$ \frac{6-4 \sqrt{2}}{6+4 \sqrt{2}} $
Given:
\( \frac{6-4 \sqrt{2}}{6+4 \sqrt{2}} \)
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,
$\frac{6-4 \sqrt{2}}{6+4 \sqrt{2}}=\frac{(6-4 \sqrt{2})(6-4 \sqrt{2})}{(6+4 \sqrt{2})(6-4 \sqrt{2})}$
$=\frac{(6-4 \sqrt{2})^{2}}{(6)^{2}-(4 \sqrt{2})^{2}}$
$=\frac{36+16 \times 2-2 \times 6 \times 4 \sqrt{2}}{36-32}$
$=\frac{36+32-48 \sqrt{2}}{4}$
$=\frac{68-48 \sqrt{2}}{4}$
$=17-12 \sqrt{2}$
Hence, $\frac{6-4 \sqrt{2}}{6+4 \sqrt{2}}=17-12 \sqrt{2}$.