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

1.	X^3 + 1/x^3 = ? What is the answer
Answer» 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 , $\: \: \: { { \bold{x {}^{3} + \dfrac{1}{x {}^{3} }= 0 }}}$ $\\ \rule{220}{2} \\$

Answer»

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

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

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