Question

5√2+4√3/5√2-4√3 rationalize the denominator

Answer 1

Given :

$\frac{5 \sqrt{2} + 4 \sqrt{3} }{5 \sqrt{2} - 4 \sqrt{3} }$

To rationalize the denominator, we have to multiply numerator and denominator by the denominator but by changing the sign.

i.e.

$= \frac{5 \sqrt{2} + 4 \sqrt{3} }{5 \sqrt{2} - 4 \sqrt{3} } \times \frac{5 \sqrt{2} + 4 \sqrt{3} }{5 \sqrt{2} + 4 \sqrt{3} } \\ \\ \\ = \frac{ {(5 \sqrt{2} + 4 \sqrt{3} )}^{2} }{(5 \sqrt{2} - 4 \sqrt{3} )(5 \sqrt{2} + 4 \sqrt{3}) }$

Using identities –

1. ${\boxed{\boxed{(a + b)^2 \ = \ a^2 + b^2 + 2ab}}}$

2. ${\boxed{\boxed{(a - b)(a + b) \ = \ a^2 - b^2}}}$


$= \frac{ {(5 \sqrt{2}) }^{2} + {(4 \sqrt{3}) }^{2} + 2 \times 5 \sqrt{2} \times 4 \sqrt{3} }{ {(5 \sqrt{2} )}^{2} - {(4 \sqrt{3}) }^{2} }$

On solving further, we get

$= \frac{50 + 48 + 40 \sqrt{6} }{50 - 48} \\ \\ \\ = \frac{98 + 40 \sqrt{6} }{2} \\ \\ \\ = \frac{2 \: (49 + 20 \sqrt{6}) }{2} \\ \\ \\ = 49 + 20 \sqrt{6}$

Answer 2

Step-by-step explanation:

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