Question

if x-1/x =9 find the value of x²+1/x² and x⁴+1/x⁴​

Answer 1

$\bf{Given}$

$\sf \to \: x - \dfrac{1}{x} = 9$

$\bf \: To \: find$

$\sf \to \: {x}^{2} + \dfrac{1}{ {x}^{2} } \\ \sf \to \: {x}^{4} + \dfrac{1}{ {x}^{4} }$

$\bf\:Now\:Take$

$\sf \to \: x - \dfrac{1}{x} = 9$

$\bf{Squaring\:on\:both\:sides}$

$\sf \to \bigg(x - \dfrac{1}{x} \bigg) ^{2} = (9) {}^{2}$

$\sf \to \: {x}^{2} + \dfrac{1}{ {x}^{2} } - 2 \times x \times \dfrac{1}{x} = 81$

$\sf \to \: {x}^{2} + \dfrac{1}{ {x}^{2} } - 2 = 81$

$\sf \to \: {x}^{2} + \dfrac{1}{ {x}^{2} } = 81 + 2$

$\sf \to \: {x}^{2} + \dfrac{1}{ {x}^{2} } = 83$

$\bf{Now \: Take }$

$\sf \to \: {x}^{2} + \dfrac{1}{ {x}^{2} } = 83$

$\bf{Squaring\:on\:both\:sides}$

$\sf \to \: \bigg( {x}^{2} + \dfrac{1}{ {x}^{2} } \bigg)^{2} = (83) {}^{2}$

$\sf \to \: ( {x}^{2} ) {}^{2} + \dfrac{1}{ ( {x}^{2} ) {}^{2} } + 2 \times {x}^{2} \times \dfrac{1}{ {x}^{2} } = 6889$

$\to\sf {x}^{4} + \dfrac{1}{ {x}^{4} } + 2 = 6889$

$\to \sf \: {x}^{4} + \dfrac{1}{ {x}^{4} } = 6889 - 2$

$\to \sf \: {x}^{4} + \dfrac{1}{ {x}^{4} } = 6887$

$\bf \: Answer$

$\sf \to \: {x}^{2} + \dfrac{1}{ {x}^{2} } = 83$

$\to \sf \: {x}^{4} + \dfrac{1}{ {x}^{4} } = 6887$

Answer 2

Answer:

$given \: : \\ x - \frac{1}{x} = 9 \\ \\ to \: find : \\ {x}^{2} + \frac{1}{ {x}^{2} } \: \: \: and \: \: \: {x}^{4} + \frac{1}{ {x}^{4} } \\ \\ now \: take : \\ x - \frac{1}{x} = 9 \\ \\ squaring \: on \: both \: side : \\ (x - \frac{1}{2} ) {}^{2} = (9) {}^{2} \\ \\ : \implies {x}^{2} + \frac{1}{ {x}^{2} } \times x \: \times \frac{1}{x} = 81 \\ \\ : \implies \: {x}^{2} + \frac{1}{ {x}^{2} } - 2 = 81 \\ \\ : \implies \: {x}^{2} + \frac{1}{ {x}^{2} } = 81 + 2 \\ \\ : \implies \: {x}^{2} + \frac{1}{ {x}^{2} } = 83 \\ \\ now \: take : \\ {x}^{2} + \frac{1}{ {x}^{2} } = 83 \\ \\ squaring \: on \: both \: side : \\ ( {x}^{2} + \frac{1}{ {x}^{2} } ) {}^{2} = (83) {}^{2} \\ \\ : \implies(x {}^{2} {)}^{2} + \frac{1}{( {x}^{2}) {}^{2} } + 2 \times {x}^{2} \times \frac{1}{ {x}^{2} } = 6889 \\ \\ : \implies {x}^{4} + \frac{1}{ {x}^{4} } + 2 = 6889 \\ \\ : \implies {x}^{4} + \frac{1}{ {x}^{4} } = 6889 - 2 \\ \\ : \implies {x}^{4} + \frac{1}{ {x}^{4} } = 6887 \\ \\ therefore : \\ : \implies \: {x}^{2} + \frac{1}{ {x}^{2} } = 83 \\ \\ : \implies \: {x}^{4} + \frac{1}{ {x}^{4} } = 6887$