Question

if a= 9-4√5, find the value of (a-1/a)²​

Answer 1

$\blue{\bold{\underline{\underline{Answer:}}}}$

$\green{\tt{\therefore{(a-\frac{1}{a})^{2}=320}}}$

$\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\green{\underline \bold{Given :}} \\ \tt: \implies a = 9 - 4 \sqrt{5} \\ \\ \red{\underline \bold{To \: Find :}} \\ \tt: \implies( a - \frac{1}{a} )^{2} =?$

• According to given question :

$\bold{As \: we \: know \: that} \\ \tt: \implies {(a - \frac{1}{a}) }^{2} = {a}^{2} + \frac{1}{ { a }^{2} } - 2 \times a \times \frac{1}{a} \\ \\ \tt: \implies {(a - \frac{1}{a}) }^{2} = {(9 - 4 \sqrt{5}) }^{2} + \frac{1}{(9 - 4 \sqrt{5} )^{2} } - 2 \\ \\ \tt: \implies {(a - \frac{1}{a}) }^{2} = {9}^{2} + {(4 \sqrt{5}) }^{2} - 72 \sqrt{5} + \frac{1}{ {9}^{2} + {(4 \sqrt{5 }) }^{2} - 72 \sqrt{5} } - 2 \\ \\ \tt: \implies {(a - \frac{1}{a}) }^{2} =81 + 80 - 72 \sqrt{5} + \frac{1}{81 + 80 - 72 \sqrt{5} } - 2$

$\tt: \implies {(a - \frac{1}{a}) }^{2} =161 - 72 \sqrt{5} + \frac{1}{161 - 72 \sqrt{5} } \times \frac{161 + 72 \sqrt{5} }{161 + 72\sqrt{5} } - 2 \\ \\ \tt: \implies {(a - \frac{1}{a}) }^{2} =161 - 72 \sqrt{5} + \frac{161 + 72\sqrt{5} }{ {161}^{2} - {(72 \sqrt{5}) }^{2} } - 2 \\ \\ \tt: \implies {(a - \frac{1}{a}) }^{2} =161 - 72 \sqrt{5} + \frac{161 + 72 \sqrt{5} }{25921 - 25920} - 2 \\ \\ \tt: \implies {(a - \frac{1}{a}) }^{2} =161 - 72 \sqrt{5} + 161 + 72 \sqrt{5} - 2 \\ \\ \tt: \implies {(a - \frac{1}{a}) }^{2} =322 - 2 \\ \\ \green{\tt: \implies {(a - \frac{1}{a}) }^{2} =320 }$

Answer 2

GiveN :-

a = 9 - 4√5

TO EvaluatE :-

$\displaystyle{(\sf{a-\frac{1}{a} })^2}$

SolutioN :-

$\displaystyle{(\sf{ a-\frac{1}{a} })^2}\\\\\displaystyle{\sf{=\{ (9-4\sqrt{5})-\frac{1}{(9-4\sqrt{5})} }\}^2}\\\\\\\displaystyle{\sf{=(9-4\sqrt{5})^2+[\frac{1}{(9-4\sqrt{5})} }]^2-2\times(9-4\sqrt{5})\times\frac{1}{(9-4\sqrt{5})} }}\\\\\\\displaystyle{\sf{=(9-4\sqrt{5})^2+[\frac{1}{(9-4\sqrt{5})} }]^2-2}}$  ....(1)

Lets find the value of (9 - 4√5)².

(9 - 4√5)² = (9)² + (4√5)² - 2 × 9 × 4√5

= 81 + (16 × 5) - 72√5

= 81 + 80 - 72√5

= 161 - 72√5

Putting this value in equation (1) :-

$\displaystyle{\sf{=161-72\sqrt{5} +\frac{1}{(161-72\sqrt{5})} -2}}\\\\\\\displaystyle{\sf{=161-72\sqrt{5} +\frac{161+72\sqrt{5} }{(161-72\sqrt{5})(161+72\sqrt{5})} -2}}\\\\\\\displaystyle{\sf{=161-72\sqrt{5} +\frac{161+72\sqrt{5} }{(161)^2-(72\sqrt{5})^2} -2}}\\\\\displaystyle{\sf{=161-72\sqrt{5} +\frac{161+72\sqrt{5} }{25921-(5184\times5)}-2}}\\\\\\\displaystyle{\sf{=161-72\sqrt{5}+\frac{161+72\sqrt{5}}{25921-25920}-2 }}$

$\displaystyle{\sf{=161-72\sqrt{5}+\frac{161+72\sqrt{5}}{1}-2 }}\\\\\\\displaystyle{\sf{=161-72\sqrt{5}+161+72\sqrt{5}-2 }}\\\\\displaystyle{\sf{=161+161-2 }}\\\\\displaystyle{\sf{=322-2 }}\\\\\Large\underline{\underline{\boxed{\boxed{\displaystyle{\sf{=320 }}}}}}$