Question

$\frac{1}{7 + 4 \sqrt{3} } + \frac{1}{2 + \sqrt{5} }$
simplify
​

Answer 1

Step-by-step explanation:

$\frac{1}{7 + 4\sqrt{3} } + \frac{1}{2 + \sqrt{5} }$

Taking 1st part, and rationalise the denominator,

$\frac{1}{7 + 4 \sqrt{3} } \times \frac{7 - 4 \sqrt{3} }{7 - 4 \sqrt{3} }$

$= \frac{7 - 4 \sqrt{3} }{(7 + 4 \sqrt{3} )(7 - 4 \sqrt{3}) }$

$= \frac{7 - 4 \sqrt{3} }{( {7)}^{2} - ( {4 \sqrt{3} )}^{2} }$

$= \frac{7 - 4 \sqrt{3} }{49 - 16 \times 3}$

$= \frac{7 - 4 \sqrt{3} }{49 - 48}$

$= \frac{7 - 4 \sqrt{3} }{1} = 7 - 4 \sqrt{3}$

Taking 2nd part, and rationalise the denominator,

$\frac{1}{2 + \sqrt{5} } \times \frac{2 - \sqrt{5} }{2 - \sqrt{5} }$

$= \frac{2 - \sqrt{5} }{(2 + \sqrt{5} )(2 - \sqrt{5} )}$

$= \frac{2 - \sqrt{5} }{( {2)}^{2} - ( { \sqrt{5}) }^{2} }$

$= \frac{2 - \sqrt{5} }{4 - 5}$

$= \frac{2 - \sqrt{5} }{ - 1}$

$= - 1(2 - \sqrt{5} ) = \sqrt{5 } - 2$

Given,

$(7 - 4 \sqrt{3} ) + ( \sqrt{5} - 2)$

$= 7 - 4 \sqrt{3} + \sqrt{5} - 2$

$= 5 - 4 \sqrt{3} + \sqrt{5}$