Question

X=1+√2 than show that {x-(1÷x)}^3=8

Answer 1

Given:-

$x = 1 + \sqrt{2}$

To Prove:-

$(x - \frac{1}{x} ) ^{3} = 8$

Solution:-

Substituting 1+√2 in place of x

that is

$(x - \frac{1}{x} ) ^{3} \\ (1 + \sqrt{2} - ( \frac{1}{1 + \sqrt{2} } )) ^{3} = 8$

Rationalize

$(1 + \sqrt{2} - ( \frac{1}{1 + \sqrt{2} } \times \frac{1 - \sqrt{2} }{1 - \sqrt{2} } )) ^{3} \\ = 8 \\ (1 + \sqrt{2} - ( \frac{1 - \sqrt{2} }{ 1 - 2} )) ^{3} \\ (1 + \sqrt{2} - ( - (1 - \sqrt{2} )) ^{3} = 8 \\ (1 + \sqrt{2} + (1 - \sqrt{2} )) ^{3} = 8 \\ (1 + \sqrt{2} + 1 - \sqrt{2} ) ^{3} = 8 \\ {2}^{3} = 8 \\ 8 = 8$

L.H.S=R.H.S

hence verified