Question

Find the value of x^4+1/x^4 , when x-1/x=4

Answer 1

$\large\underline{\sf{Solution-}}$

Given that

$\rm :\longmapsto\:x - \dfrac{1}{x} = 4$

On squaring both sides, we get

$\rm :\longmapsto\: {\bigg[x - \dfrac{1}{x} \bigg]}^{2} = {4}^{2}$

$\rm :\longmapsto\: {x}^{2} + \dfrac{1}{ {x}^{2} } - 2 \times x \times \dfrac{1}{x} = 16$

$\rm :\longmapsto\: {x}^{2} + \dfrac{1}{ {x}^{2} } - 2 = 16$

$\rm :\longmapsto\: {x}^{2} + \dfrac{1}{ {x}^{2} } = 16 + 2$

$\rm :\longmapsto\: {x}^{2} + \dfrac{1}{ {x}^{2} } = 18$

On squaring both sides, we get

$\rm :\longmapsto\: {\bigg[ {x}^{2} + \dfrac{1}{ {x}^{2} } \bigg]}^{2} = {18}^{2}$

$\rm :\longmapsto\: {x}^{4} + \dfrac{1}{ {x}^{4} } + 2 \times {x}^{2} \times \dfrac{1}{ {x}^{2} } = 324$

$\rm :\longmapsto\: {x}^{4} + \dfrac{1}{ {x}^{4} } + 2 = 324$

$\rm :\longmapsto\: {x}^{4} + \dfrac{1}{ {x}^{4} } = 324 - 2$

$\rm :\longmapsto\: {x}^{4} + \dfrac{1}{ {x}^{4} } = 322$

Hence,

$\purple{\rm :\longmapsto\: \boxed{\tt{ \: {x}^{4} + \dfrac{1}{ {x}^{4} } = 322}}}$

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Formula Used

$\red{\rm :\longmapsto\:\boxed{\tt{ \: {(x - y)}^{2} = {x}^{2} - 2xy + {y}^{2} \: }}}$

$\red{\rm :\longmapsto\:\boxed{\tt{ \: {(x + y)}^{2} = {x}^{2} + 2xy + {y}^{2} \: }}}$

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More Identities to Know

$\pink{\rm :\longmapsto\:\boxed{\tt{ \: {(x + y)}^{3} = {x}^{3} + 3xy(x + y) + {y}^{3} \: }}}$

$\pink{\rm :\longmapsto\:\boxed{\tt{ \: {(x - y)}^{3} = {x}^{3} - 3xy(x - y) - {y}^{3} \: }}}$

$\green{\rm :\longmapsto\:\boxed{\tt{ \: {(x + y)}^{2} + {(x - y)}^{2} = 2( {x}^{2} + {y}^{2}) \: }}}$

$\green{\rm :\longmapsto\:\boxed{\tt{ \: {(x + y)}^{2} - {(x - y)}^{2} = 4xy \: }}}$