Question

If $a = 9 - 4 \sqrt{5}$
,find the value of $a \frac{1}{a}$

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Answer 1

$Given \: Question \: Is \: \: \\ \\ a = 9 - 4 \sqrt{5} \\ \\ Answer \: \: \\ \\ \frac{1}{a} = \frac{1}{9 - 4 \sqrt{5} } \\ \\ Multiply \: \: Numerator \: \: by \: 9 + 4 \sqrt{5} \\ \\ \frac{1}{a} = \frac{1}{9 - 4 \sqrt{5} } \times \frac{9 + 4 \sqrt{5} }{9 + 4 \sqrt{5} } \\ \\ \frac{1}{a} = \frac{9 + 4 \sqrt{5} }{9 {}^{2} - (4 \sqrt{5} ) {}^{2} } \\ \\ \frac{1}{a} = \frac{9 + 4 \sqrt{5} }{81 - 80} \\ \\ \frac{1}{a} = \frac{ 9 + 4 \sqrt{5} }{1} \\ \\ \frac{1}{a} = 9 + 4 \sqrt{5} \\ \\ so \: \: \: a + \frac{1}{a} = 9 - 4 \sqrt{5} + 9 + 4 \sqrt{5} \\ \\ a + \frac{1}{a} = 18 \\ \\ Therefore \: \: a + \frac{1}{a} = 18 \\ \\ Note \: \\ \\ 1) \: \: (a + b)(a - b) = a {}^{2} - b {}^{2} \\ \\ 2) \: \: \: a \frac{1}{a} = a + \frac{1}{a}$

Answer 2

Here is the answer...!!!

Refer to the attachment...!!!