Question

If x=3-2√2, find x^3+1/x^3​

Answer 1

$x = 3 - 2 \sqrt{2} \\ \frac{1}{x} = \frac{1}{3 - 2 \sqrt{2} } \\ \: \: \: \: \: = \frac{1}{3 - 2 \sqrt{2} } \times \frac{3 + 2 \sqrt{2} }{3 + 2 \sqrt{2} } \\ \: \: \: \: \: = \frac{3 + 2 \sqrt{2} }{( {3)}^{2} - (2 \sqrt{2} ) ^{2} } \\ \: \: \: \: \: = \frac{3 + 2 \sqrt{2} }{9 - 8} \\ \: \: \: \: \: = 3 + 2\sqrt{2} \\ = > x + \frac{1}{x} =3 - 2\sqrt{2} + 3 + 2 \sqrt{2} = 6 \\ = > {x}^{3} + \frac{1}{ {x}^{3} } = (x + \frac{1}{x} ) ^{3} - 3(x + \frac{1}{x} ) \\ = > {x}^{3} + \frac{1}{ {x}^{3} } =(6) ^{3} - 3(6) \\ = > {x}^{3} + \frac{1}{ {x}^{3} } =216 - 18 \\ = > {x}^{3} + \frac{1}{ {x}^{3} } =198$