If-x-1-x-6-then-x-3-1-x-3-

Question Number 155259 by Fridunatjan08 last updated on 27/Sep/21

I f : x + \frac{1}{x} = 6 t h e n x^{3} + \frac{1}{x^{3}} = ?

Commented by puissant last updated on 27/Sep/21

{(x + \frac{1}{x})}^{3} = 6^{3} = 216

{(x + \frac{1}{x})}^{3} = x^{3} + 3 x^{2} \times \frac{1}{x} + 3 x \times \frac{1}{x^{2}} + \frac{1}{x^{3}}

= x^{3} + \frac{1}{x^{3}} + 3 (x + \frac{1}{x}) = 216

\Rightarrow x^{3} + \frac{1}{x^{3}} = 216 - 3 \times 6 = 198 . .

Commented by mathdanisur last updated on 27/Sep/21

x + \frac{1}{x} = a \Rightarrow x^{3} + \frac{1}{x^{3}} = a^{3} - 3 a \Rightarrow 198

Answered by Rasheed.Sindhi last updated on 28/Sep/21

A n A lternate W ay

\begin{matrix} a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2} \end{matrix}

x^{3} + \frac{1}{x^{3}} = (x + \frac{1}{x}) (x^{2} - 1 + \frac{1}{x^{2}})

= (x + \frac{1}{x}) ({(x + \frac{1}{x})}^{2} - 2 - 1)

= (6) (6^{2} - 3) = 6 \times 33 = 198

Commented by peter frank last updated on 28/Sep/21

great

Commented by Rasheed.Sindhi last updated on 28/Sep/21

Thank you!