Question

Rationalize
$6 - 4 \sqrt{2} \div 6 + 4 \sqrt{2}$

Answer 1

Heya there !!!
Here is the answer you were looking for:
$\frac{6 - 4 \sqrt{2} }{6 + 4 \sqrt{2} }$

On rationalizing the denominator we get,

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

Using the identities:

${(x + y)}^{2} = {x}^{2} + {y}^{2} - 2xy \\ ( x + y)(x - y) = {x}^{2} - {y}^{2}$


$= \frac{ {(6)}^{2} + {(4 \sqrt{2} )}^{2} - 2(6)(4 \sqrt{2}) }{ {(6)}^{2} - {(4 \sqrt{2} )}^{2} } \\ \\ = \frac{36 + 32 - 48 \sqrt{2} }{36 - 32} \\ \\ = \frac{68 - 48 \sqrt{2} }{4} \\ \\ = \frac{4(17 - 12 \sqrt{2} )}{4} \\ \\ = 17 - 12 \sqrt{2}$


Hope this helps!!!

Feel free to ask in comment section if you have any doubt regarding to my answer...

@Mahak24

Thanks...
☺☺