Question

Find the values of a and b in the following
$\frac{7 + \sqrt{ 5} }{7 - \sqrt{5} } - \frac{7 - \sqrt{5} }{7 + \sqrt{5} } = a + \frac{7}{11} \sqrt{5} \: b$
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Answer 1

*Solution* :-

Step-by-step explanation:

★Question★

$\\ \frac{7 + \sqrt{5} }{7 - \sqrt{5} } - \frac{7 - \sqrt{5} }{7 + \sqrt{5} }$

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†Answer†

$\ \frac{7 + \sqrt{5} }{7 - \sqrt{5} } - \frac{7 - \sqrt{5} }{7 + \sqrt{5} }$

Rationalising the denominator, we get

$\frac{7 + \sqrt{5} }{7 - \sqrt{5} } \times \frac{7 + \sqrt{5} }{7 + \sqrt{5} } - \frac{7 - \sqrt{5 } }{7 + \sqrt{5 } } \times \frac{7 - \sqrt{5} }{7 - \sqrt{5} }$

$= \frac{(7 + \sqrt{5) ^{2} } }{7 ^{2} - ( \sqrt{5) ^{2} } } - \frac{(7 - \sqrt{5) ^{2} } }{7 ^{2} - ( \sqrt{5}) ^{2} }$

$\frac{7^{2} + ( \sqrt{5})^{2}+ 2 \times 7 \ \sqrt{5} }{49 - 5} - \ \frac{7^{2} + ( { \sqrt{5} )^{2} - 2 \times 7 \sqrt{5} } }{49 - 5}$

$\frac{49 + 5 + 14 \sqrt{5} }{44} - \frac{49 + 5 - 14 \sqrt{5} }{44}$

$= \frac{54 + 14 \sqrt{5 - 54 + 14 \sqrt{5} } }{44}$

$= \frac{28 \sqrt{5} }{44}$

$= \frac{7 \sqrt{5} }{11}$

$= \frac{0 + 7 \sqrt{5} }{11}$

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Hence,

: ⟹

$\frac{0 + 7 \sqrt{5} }{11 } = \frac{a + 7 \sqrt{5}b }{11}$

: ⟹

a = 0,b =1