Question

x=2+root3 show that x^3 +1/x^3 =52

Answer 1

since
x=2+√3,

then
x=2-√3,

then

x³+1/x³=(2+√3)³+(2-√3)³,

=[8+3√3+3×2×√3(2+√3)]+[8-3√3-3×2×√3(2-√3)],

=[8+3√3+12√3+18]+[8-3√3-12√3+18],

=8+3√3+12√3+18+8-3√3-12√3+18,

=8+8+18+18,

=52✓✓✓✓✓

Answer 2

$given \: \: x = 2 + \sqrt{3} \\ so \: \: \frac{1}{x} = \frac{1}{2 + \sqrt{3}} = \frac{2 - \sqrt{3}}{(2 + \sqrt{3})(2 - \sqrt{3})} \\ = \frac{2 - \sqrt{3}}{4 - 3} = 2 - \sqrt{3} \\ \\ therefore \: \: x + \frac{1}{x} \\ = (2 + \sqrt{3}) + (2 - \sqrt{3}) = 4 \\ \\ {x}^{3} + \frac{1}{ {x}^{3} } = {(x + \frac{1}{x} )}^{3} - 3(x + \frac{1}{x} ) \\ = {4}^{3} - 3 \times 4 = 64 - 12 = 52$