Question

X-1/x=3+2root2. Find x^3-1/x^3

Answer 1

$=> x - \frac{1}{x} = 3 + 2 \sqrt{2} \\ = > {(x - \frac{1}{x} })^{3} = {(3 + 2 \sqrt{2} })^{3} \\ => {x}^{3} - \frac{1}{ {x}^{3} } - 3(x - \frac{1}{x} ) =27 + 16 \sqrt{2} +54 \sqrt{2} + 72 \\ = > {x}^{3} - \frac{1}{ {x}^{3} } - 3(3 + 2 \sqrt{2)} = 99 + 70 \sqrt{2} \\ = > {x}^{3} - \frac{1}{ {x}^{3} } = 99 + 70 \sqrt{2} + 9 + 6 \sqrt{2} \\ = > {x}^{3} - \frac{1}{ {x}^{3} } = 108 + 76 \sqrt{2}$