Question

If (x+1/x)2=3,find x3+1/x3​

Answer 1

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

Cubing on both sides, we get

$(x + \frac{1}{x} ) ^{3} = ±3 \sqrt{3} \\ {x}^{3} + \frac{1}{ {x}^{3} } + 3x + \frac{3}{x} = ±3 \sqrt{3} \\ {x}^{3} + \frac{1}{ {x}^{3} } + 3(x + \frac{1}{x} ) = ±3 \sqrt{3} \\ {x}^{3} + \frac{1}{ {x}^{3} } + 3( ± \sqrt{3} ) = ±3 \sqrt{3} \\ {x}^{3} + \frac{1}{x ^{3} } ± 3\sqrt{3} = ±3 \sqrt{3} \\ {x}^{3} + \frac{1}{ {x}^{3} } = 0$