Simplify: $\frac{6-4 \sqrt{2}}{6+4 \sqrt{2}}$.

Given: $\frac{6-4 \sqrt{2}}{6+4 \sqrt{2}}$.

To do: To simplify $\frac{6-4 \sqrt{2}}{6+4 \sqrt{2}}$.

Solution:

$\frac{6-4 \sqrt{2}}{6+4 \sqrt{2}}$

On multiplying by $( 6-4\sqrt{2})$ both the numerator and denominator

$=\frac{6-4 \sqrt{2}}{6+4 \sqrt{2}}\times\frac{6-4 \sqrt{2}}{6-4 \sqrt{2}}$

$=\frac{( 6-4 \sqrt{2})^2}{6^2-( 4\sqrt{2})^2}$            [$\because ( a-b)( a-b)=( a-b)^2$ and $( a-b)( a+b)=a^2-b^2$]

$=\frac{6^2-2\times6\times4\sqrt{2}+( 4\sqrt{2})^2}{36-4\times4\times2}$

$=\frac{36-48\sqrt{2}+32}{36-32}$

$=\frac{68-48\sqrt{2}}{4}$

$=\frac{4( 17-12\sqrt{2})}{4}$

$=17-12\sqrt{2}$

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Updated on: 10-Oct-2022

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