Question

x+1/x=p then what is x^6+1/x^6 equal to​

Answer 1

$\huge{\underline{\underline{\purple{♡Solution→}}}}$

Given →

$x + \frac{1}{x} = p$

To Find →

${x}^{6} + \frac{1}{ {x}^{6} } = \: ?$

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By squaring of given Equation →

${(x + \frac{1}{x} )}^{2} = {p}^{2}$

${x}^{2} + \frac{1}{ {x}^{2} } + 2 \times x \times \frac{1}{x} = {p}^{2}$

${x}^{2} + \frac{1}{ {x}^{2} } + 2 = {p}^{2} \\ {x}^{2} + \frac{1}{ {x}^{2} } = {p}^{2} - 2 - - - - - (i)$

Now Cubing of this equation →

${( {x}^{2} + \frac{1}{ {x}^{2} } ) }^{3} = {( {p}^{2} - 2) }^{3}$

$( { {x}^{2}) }^{3} + ( { \frac{1}{ {x}^{2} }) }^{3} + 3 \times {x}^{2} \times \frac{1}{ {x}^{2} } ( {x}^{2} + \frac{1}{ {x}^{2} } ) = ( { {p}^{2} - 2)}^{3}$

${x}^{6} + \frac{1}{ {x}^{6} } + 3( {x}^{2} + \frac{1}{ {x}^{2} } ) = ( { {p}^{2} - 2)}^{3}$

Now put the value of ${x}^{2} + \frac{1}{ {x}^{2} }$ From (i)

${x}^{6} + \frac{1}{ {x}^{6} } + 3( {p}^{2} - 2) = {( {p}^{2} - 2) }^{3}$

${x}^{6} + \frac{1}{ {x}^{6} } + 3 {p}^{2} - 6 = {p}^{6} - 8 - 3 \times {p}^{2} \times 2( {p}^{2} - 2) \\ {x}^{6} + \frac{1}{ {x}^{6} } + 3 {p}^{2} - 6 = {p}^{6} - 8 - 6 {p}^{2} ( {p}^{2} - 2) \\ {x}^{6} + \frac{1}{ {x}^{6} } + 3 {p}^{2} - 6 = {p}^{6} - 8 - 6 {p}^{4} + 12 {p}^{2} \\ {x}^{6} + \frac{1}{ {x}^{6} } = {p}^{6} - 8 - 6 {p}^{4} + 12 {p}^{2} - 3 {p}^{2} + 6 \\ {x}^{6} + \frac{1}{ {x}^{6} } = {p}^{6} - 6 {p}^{4} + 9 {p}^{2} - 2$

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