Question

if x-1/x = 3+2√2,find the value of x3-1/x3​

Answer 1

Answer:

given

$x - \frac{1}{x} = 3 + 2 \sqrt{2}$

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

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

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

$\: \: \: \: \: \: \: = 3 + 2 \sqrt{2} ((3 + 2 \sqrt{2} ) {}^{2} + 3)$

$\: \: \: \: \: \: \: \: \: \: = 3 + 2 \sqrt{2} (9 + 8 + 12 \sqrt{2} + 3$

$\: \: \: \: \: \: \: \: \: = 63 + 36 \sqrt{2} + 42 \sqrt{2} + 24 \times 2$

$\: \: \: \: \: \: \: \: \: = 63 + 48 + 78 \sqrt{2}$

${x}^{3} - \frac{1}{ {x}^{3} } = 111 + 78 \sqrt{2}$