Question

X+1/x=root 5 , find the value of x^3 +1/x^3

Answer 1

Given,

${x}+\frac{1}{x}=\sqrt{5}$

Expanding :- ${({x}+\frac{1}{x})}^{3}$

${({x}+\frac{1}{x})}^{3} = {x}^{3} +\frac{1}{x}^{3} + 3({x}+\frac{1}{x})$

$\sqrt{5}^{3} = {x}^{3} +\frac{1}{x}^{3} + {3}\sqrt{5}$

${5}\sqrt{5} - {3}\sqrt{5}= {x}^{3} +\frac{1}{x}^{3}$

${5}\sqrt{5} - {3}\sqrt{5}= {x}^{3} +\frac{1}{x}^{3}$

$\huge{\boxed{\red{\bf{Answer:-}}}}$

${x}^{3} +\frac{1}{x}^{3} ={2}\sqrt{5}$

Answer 2

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Question :-

$if\: x + \frac{1}{x} = \sqrt 5$

then find :-

$x^3 + \frac{1}{x^3}$

formulae used :-

$( x + \frac{1}{x})^3 = x^3 + \frac{1}{x^3} + 3 ( x + \frac{1}{x})$

putting values :-

$( x + \frac{1}{x})^3 = x^3 + \frac{1}{x^3} + 3 ( x + \frac{1}{x})$

$(\sqrt5)^3 = x^3 + \frac{1}{x^3} + 3(\sqrt5)$

$5\sqrt5= x^3 + \frac{1}{x^3} + 3\sqrt 5$

$5\sqrt 5 - 3 \sqrt 3= x^3 + \frac{1}{x^3}$

$x^3 + \frac{1}{x^3} =2\sqrt 5$

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