Question

if x-1/X = 3 , find the value of x²+1/x² and x⁴+1/x⁴​

Answer 1

$\bf{Given}$

$\rm \to \: x - \dfrac{1}{x} = 3$

$\bf{To \: Find }$

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

$\bf{Now \: \: Take }$

$\rm \to \: x - \dfrac{1}{x} = 3$

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

$\rm \to \: \bigg(x - \dfrac{1}{x} \bigg) ^{2} = (3)^{2}$

$\bf{We \: Get}$

$\to \rm\: {x}^{2} + \dfrac{1}{ {x}^{2} } - 2 \times x \times \dfrac{1}{x} = 9$

$\rm \to \: {x}^{2} + \dfrac{1}{ {x}^{2} } - 2 = 9$

$\rm \to \: {x}^{2} + \dfrac{1}{ {x}^{2} } = 9 + 2$

$\rm \to \: {x}^{2} + \dfrac{1}{ {x}^{2} } = 11$

$\bf{Now \: Take }$

$\rm \to \: {x}^{2} + \dfrac{1}{ {x}^{2} } = 11$

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

$\rm \to \: \bigg( {x}^{2} + \dfrac{1}{ {x}^{2} } \bigg) ^{2} = (11)^{2}$

$\rm \to \: {x}^{4} + \dfrac{1}{ {x}^{4} } + 2 \times \dfrac{1}{ {x}^{2} } \times {x}^{2} = 121$

$\to \rm \: {x}^{4} + \dfrac{1}{ {x}^{4} } + 2 = 121$

$\to \rm \: {x}^{4} + \dfrac{1}{ {x}^{4} } = 121 - 2$

$\to \rm \: {x}^{4} + \dfrac{1}{ {x}^{4} } = 119$

$\bf{Answer}$

$\rm \to \: {x}^{2} + \dfrac{1}{ {x}^{2} } = 11$

$\to \rm \: {x}^{4} + \dfrac{1}{ {x}^{4} } = 119$