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, $\frac{6-4 \sqrt{2}}{6+4 \sqrt{2}}=17-12 \sqrt{2}$.

Tutorialspoint

Updated on: 10-Oct-2022

695 Views

Kickstart Your Career

Get certified by completing the course

Get Started