Question

if a is equal to 1 minus root 2 then find the value of a minus one upon a raise to the power 3

Answer 1

$a = 1 - \sqrt{2} \\ {a}^{2} = 1 + 2 - 2 \sqrt{2} = 3 - 2 \sqrt{2} \\ {a}^{3} = (1 - \sqrt{2})(3 - 2 \sqrt{2} ) = 3 - 3 \sqrt{2} - 2 \sqrt{2 } + 4 = 7 - 5 \sqrt{2} \\ {a}^{4} = 9 + 8 - 12 \sqrt{2} = 17 - 12 \sqrt{2} \\ a - \frac{1}{ {a}^{3} } = \frac{ {a}^{4} - 1}{ {a}^{3} } = \frac{17 - 12 \sqrt{2} - 1}{7 - 5 \sqrt{2} } \\ = \frac{16 - 12 \sqrt{2} }{7 - 5 \sqrt{2} } = \frac{(16 - 12 \sqrt{2} )(7 + 5 \sqrt{2}) }{(7 + 5 \sqrt{2} )(7 - 5 \sqrt{2} )} \\ = \frac{112 - 84 \sqrt{2} + 80 \sqrt{2} - 120 }{49 - 50} \\ = \frac{ - 8 - 4 \sqrt{2} }{ - 1} = 8 + 4 \sqrt{2}$