Question

solve these please. ​

Answer 1

GIVEN :–

$\\ \sf \: { \huge{.}} \: \: {x}^{4} + \dfrac{1}{ {x}^{4} } = 2207$

TO FIND :–

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

SOLUTION :–

$\\ \sf \implies {x}^{4} + \dfrac{1}{ {x}^{4} } = 2207$

• Using identity –

$\\ \sf \longrightarrow \: {(a + b)}^{2} = {a}^{2} + {b}^{2} + 2ab$

• We should write this as –

$\\ \sf \implies \left[{x}^{2} + \dfrac{1}{ {x}^{2} } \right]^{2} - 2( {x}^{2}) \left( \dfrac{1}{ {x}^{2} } \right) = 2207$

$\\ \sf \implies \left[{x}^{2} + \dfrac{1}{ {x}^{2} } \right]^{2} - 2= 2207$

$\\ \sf \implies \left[{x}^{2} + \dfrac{1}{ {x}^{2} } \right]^{2}= 2207 + 2$

$\\ \sf \implies \left[{x}^{2} + \dfrac{1}{ {x}^{2} } \right]^{2}= 2209$

$\\ \sf \implies {x}^{2} + \dfrac{1}{ {x}^{2} } = \sqrt{2209}$

$\\ \sf \implies {x}^{2} + \dfrac{1}{ {x}^{2} } = \pm \sqrt{2209}$

$\\ \sf \implies {x}^{2} + \dfrac{1}{ {x}^{2} } = \pm 47$

• Here (-ve) value is not possible , So that –

$\\ \sf \implies {x}^{2} + \dfrac{1}{ {x}^{2} } = 47$

• Again using identity –

$\\ \sf \longrightarrow \: {(a + b)}^{2} = {a}^{2} + {b}^{2} + 2ab$

• So that –

$\\ \sf \implies \left({x} + \dfrac{1}{ {x} } \right)^{2} - 2(x) \left( \dfrac{1}{x} \right) = 47$

$\\ \sf \implies \left({x} + \dfrac{1}{ {x} } \right)^{2} - 2 = 47$

$\\ \sf \implies \left({x} + \dfrac{1}{ {x} } \right)^{2} = 47 + 2$

$\\ \sf \implies \left({x} + \dfrac{1}{ {x} } \right)^{2} = 49$

$\\ \sf \implies {x} + \dfrac{1}{ {x} } = \pm \sqrt{49}$

$\\ \sf \implies {x} + \dfrac{1}{ {x} } = \pm 7$

• Now using identity –

$\\ \sf \longrightarrow \: {(a + b)}^{3} = {a}^{3} + {b}^{3} + 3ab(a + b)$

• Take positive(+) sign :–

$\\ \sf \implies \left( {x} + \dfrac{1}{ {x} } \right)^{3} =( 7)^{3}$

$\\ \sf \implies {x}^{3} + \left( \dfrac{1}{ {x} } \right)^{3} + 3(x) \left( \frac{1}{x} \right) \left(x + \dfrac{1}{x} \right) =(7)^{3}$

$\\ \sf \implies {x}^{3} + \left( \dfrac{1}{ {x} } \right)^{3} + 3(7) =(7)^{3}$

$\\ \sf \implies {x}^{3} + \left( \dfrac{1}{ {x} } \right)^{3} + 21 =343$

$\\ \sf \implies {x}^{3} + \left( \dfrac{1}{ {x} } \right)^{3} =343 - 21$

$\\ \sf \implies {x}^{3} + \left( \dfrac{1}{ {x} } \right)^{3} =322$

$\\ \implies \large{ \boxed{ \sf{x}^{3} + \dfrac{1}{ {x} ^{3}} =322 }}$

• Take negative(-) sign :–

$\\ \sf \implies \left( {x} + \dfrac{1}{ {x} } \right)^{3} =( - 7)^{3}$

$\\ \sf \implies {x}^{3} + \left( \dfrac{1}{ {x} } \right)^{3} + 3(x) \left( \frac{1}{x} \right) \left(x + \dfrac{1}{x} \right) =( - 7)^{3}$

$\\ \sf \implies {x}^{3} + \left( \dfrac{1}{ {x} } \right)^{3} + 3( - 7) =( - 7)^{3}$

$\\ \sf \implies {x}^{3} + \left( \dfrac{1}{ {x} } \right)^{3} - 21 = - 343$

$\\ \sf \implies {x}^{3} + \left( \dfrac{1}{ {x} } \right)^{3} = - 343 + 21$

$\\ \sf \implies {x}^{3} + \left( \dfrac{1}{ {x} } \right)^{3} = - 322$

$\\ \implies \large{ \boxed{ \sf{x}^{3} + \dfrac{1}{ {x} ^{3}} = - 322 }}$

Answer 2

$\sf{\color{purple}{Answer:-}}$

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$\sf{\color{grey}{Given:-}}$

$\sf {x}^{4} + \frac{1}{ {x}^{4} } = 2207$

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$\sf{\color{grey}{To~Find:-}}$

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

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$\sf{\color{grey}{Solution:-}}$

$\sf ({ {x}^{2} + \frac{1}{ {x}^{2} } })^{2} = {x}^{4} + \frac{1}{ {x}^{4} } + 2 \times {x}^{2} \times \frac{1}{ {x}^{2} } \\↦ \sf ({ {x}^{2} + \frac{1}{ {x}^{2} } })^{2} = {x}^{4} + \frac{1}{ {x}^{4} } + 2 \\↦ \sf ({ {x}^{2} + \frac{1}{ {x}^{2} } })^{2} = 2207 + 2 \\↦ \sf ({ {x}^{2} + \frac{1}{ {x}^{2} } })^{2} = 2209 \\ ↦\sf { {x}^{2} + \frac{1}{ {x}^{2} } } = \sqrt{2209} \\ ↦\sf { {x}^{2} + \frac{1}{ {x}^{2} } } = 47$

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$\sf (x + \frac{1}{x} ) {}^{2} = {x}^{2} + \frac{1}{ {x}^{2} } + 2 \times x \times \frac{1}{x} \\⇝ \sf (x + \frac{1}{x} ) {}^{2} = {x}^{2} + \frac{1}{ {x}^{2} } + 2 \\⇝ \sf (x + \frac{1}{x} ) {}^{2} = 47 + 2 \\ ⇝\sf (x + \frac{1}{x} ) {}^{2} = 49 \\ ⇝\sf x + \frac{1}{x}= \sqrt{49} \\⇝ \sf x + \frac{1}{x} = 7$

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$\sf (x + \frac{1}{x} ) {}^{3} = {x}^{3} + \frac{1}{ {x}^{3} } + 3 \times x \frac{1}{x} (x + \frac{1}{x} ) \\⇢ \sf ( x + \frac{1}{x} ) {}^{3} = {x}^{3} + \frac{1}{ {x}^{3} } + 3(x + \frac{1}{x} ) \\ ⇢\sf {7}^{3} = {x}^{3} + \frac{1}{ {x}^{3} } + 3(7) \\ ⇢ \sf 343 = {x}^{3} + \frac{1}{ {x}^{3} } + 21 \\⇢ \sf {x}^{3} + \frac{1}{ {x}^{3} } = 343 - 21 \\⇢ \sf {x}^{3} + \frac{1}{ {x}^{3} } = 322$

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$\sf{\color{purple}{Final~Answer:-}}$

$\sf {x}^{3} + \frac{1}{ {x}^{3} } = 322$