Question

if=3+under root 8 find x^4+1/x^4

Answer 1

x = 3 + √8

$\frac{1}{x} = \frac{1}{3 + \sqrt{8} }$

By Rationalization,

$\frac{1}{x} = \frac{1}{3 + \sqrt{8} } \times \frac{3 - \sqrt{8} }{3 - \sqrt{8} } \\ \\ \\ => \frac{1}{x} = \frac{3 - \sqrt{8} }{ {(3)}^{2} - {( \sqrt{8} )}^{2} } = \frac{3 - \sqrt{8} }{9 - 8} = \frac{3 - \sqrt{8} }{1} = 3 - \sqrt{8}$

Then,

$x + \frac{1}{x} = > 3 + \sqrt{8} + 3 - \sqrt{8} = > 3 + 3 = 6$

Square on both sides,

${( x + \frac{1}{x} )}^{2} = {6}^{2} \\ \\ \\ {x}^{2} + \frac{1}{ {x}^{2} } + 2(x \times \frac{1}{x} ) = 36 \\ \\ {x}^{2} + \frac{1}{ {x}^{2} } + 2 = 36 \\ \\ \\ {x}^{2} + \frac{1}{ {x}^{2} } = 36 - 2 = > 34$

Again square on both sides,

$( {x}^{2} + \frac{1}{ {x}^{2} } ) ^{2} = {34}^{2} \\ \\ => {x}^{4} + \frac{1}{ {x}^{4} } + 2 = 1156 \\ \\ \\ {x}^{4} + \frac{1}{ {x}^{4} } = 1156 - 2 = 1154$