Question

if (x^2)+(1/×^2)=51 find ((×)-(1/x) and (x)^3-(1/x)^3

Answer 1

$x {}^{2} + \frac{1}{x {}^{2} } = 51 \\ (x - \frac{1}{x} ) {}^{2} = x {}^{2} + \frac{1}{x {}^{2} } - 2 \times x \times \frac{1}{x} \\ = x {}^{2} + \frac{1}{x {}^{2} } - 2 \\ = 51 - 2 \\ = 49 \\ (x - \frac{1}{x} ) {}^{2} = 49 \\ x - \frac{1}{x} = \sqrt{49} \\ x - \frac{1}{x} = plus \: minus( + - )\: 7$
$x {}^{3} - \frac{1}{x {}^{3} } = (x - \frac{1}{x} ) {}^{ {}^{3 } } + 3 x \times \frac{1}{x} (x - \frac{1}{x} ) \\ = (7) {}^{3} + 3 \times 7 \\ = 343 + 21 \\ = 364$
So,
$x - \frac{1}{x} =( + - 7) \\ x {}^{3} - \frac{1}{x {}^{3} } = 364$