Question

Evaluate (x-1/x) (x+1/x) (x²+1/x²) (x^4+1/x^4)

Answer 1

$\large\underline\blue{\bold{Given \:  Question :-  }}$

$\bf \:Evaluate \: \bigg(x - \dfrac{1}{x} \bigg)\bigg( x + \dfrac{1}{x} \bigg)\bigg( {x}^{2} + \dfrac{1}{ {x}^{2} } \bigg)\bigg( {x}^{4} + \dfrac{1}{ {x}^{4} }$

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$\huge{AηsωeR} ✍$

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$\begin{gathered}\Large{\bold{\pink{\underline{Formula \: Used \::}}}} \end{gathered}$

$\sf \:  ⟼(1). \: (x + y)(x - y) = {x}^{2} - {y}^{2}$

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$\large\underline\purple{\bold{Solution :- }}$

$\sf \:  \bigg(x - \dfrac{1}{x} \bigg)\bigg( x + \dfrac{1}{x} \bigg)\bigg( {x}^{2} + \dfrac{1}{ {x}^{2} } \bigg)\bigg( {x}^{4} + \dfrac{1}{ {x}^{4} } \bigg)$

$\sf \:  = \bigg( {x}^{2} - \dfrac{1}{ {x}^{2} } \bigg)\bigg( {x}^{2} + \dfrac{1}{ {x}^{2} } \bigg)\bigg( {x}^{4} + \dfrac{1}{ {x}^{4} } \bigg)$

☆[using identity in first two brackets]

$\sf \:  = \bigg( ({ {x}^{2} })^{2} - \dfrac{1}{( { {x}^{2} })^{2} } \bigg)\bigg( {x}^{4} + \dfrac{1}{ {x}^{4} } \bigg)$

$\sf \:  = \bigg( {x}^{4} - \dfrac{1}{ {x}^{4} } \bigg)\bigg( {x}^{4} + \dfrac{1}{ {x}^{4} } \bigg)$

$\sf \:  = \bigg( ({ {x}^{4} })^{2} - \dfrac{1}{( { {x}^{4} })^{2} } \bigg)$

$\sf \:  = {x}^{8} - \dfrac{1}{ {x}^{8} }$

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$\large \red{\bf \: ⟼ Explore \: more } ✍$

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

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

$\sf \:  ⟼(a + b)(a - b) = {a}^{2} - {b}^{2}$

$\sf \:  ⟼(a + b + c)^{2} = {a}^{2} + {b}^{2} + {c}^{2} + 2ab + 2bc + 2ca</p><p>$

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

$\sf \:  ⟼a ^{3} + {b}^{3} = (a + b)(a ^{2} + {b}^{2} - ab) \\ \sf \:  ⟼a ^{3} - {b}^{3} = (a - b)(a ^{2} + {b}^{2} + ab)</p><p>$

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