Question

x^3 + 1/x^3 = ?
What is the answer

Answer 1

Question :–

• If $\: \: { \bold{x + \dfrac{1}{x} = \sqrt{3} }} \: \:$ , then find $\: \: { \bold{x {}^{3} + \dfrac{1}{x {}^{3} } = ? }} \: \:$

ANSWER :–

GIVEN :–

$\\ \: \: { \huge{.}} \: \: { \bold{x + \dfrac{1}{x} = \sqrt{3} }} \: \:$

TO FIND :–

$\: \: { \huge{.}} \: \: { \bold{x {}^{3} + \dfrac{1}{x {}^{3} } = ? }} \: \:$

SOLUTION :–

$\\ \: \implies \: { \bold{x + \dfrac{1}{x} = \sqrt{3} }} \: \:$

• Take cube on both side –

$\\ \: \implies \: { (\bold{x + \dfrac{1}{x} ) {}^{3} = (\sqrt{3} ) {}^{3} }} \: \:$

• Using identity –

$\\ \: \longrightarrow \: \large { \boxed{ (\bold{a+ b) {}^{3} = { a }^{3} + {b}^{3} + 3ab(a + b)}}} \: \:$

• So that –

$\\ \: \implies \: { \bold{(x ) {}^{3} + (\dfrac{1}{x}) {}^{3} + 3(x)( \dfrac{1}{x} ) [x + \dfrac{1}{x} ] = ( \sqrt{3} ) {}^{3} }}$

$\\ \: \implies \: { \bold{(x ) {}^{3} + (\dfrac{1}{x}) {}^{3} + 3(x + \dfrac{1}{x})= 3 \sqrt{3} }}$

$\\ \: \implies \: { \bold{(x ) {}^{3} + (\dfrac{1}{x}) {}^{3} + 3( \sqrt{3} )= 3 \sqrt{3} }}$

$\\ \: \implies \: { \bold{x {}^{3} + \dfrac{1}{x {}^{3} }= 3 \sqrt{3} - 3 \sqrt{3} }}$

$\\ \: \implies \large{ \boxed { \bold{x {}^{3} + \dfrac{1}{x {}^{3} }= 0 }}}$

• Hence , $\: \: \: { { \bold{x {}^{3} + \dfrac{1}{x {}^{3} }= 0 }}}$

$\\ \rule{220}{2}$