Question

if x=2+√3 , y=2-√3 find x^-3 + y^-3​

Answer 1

Answer:

$xy = (2 + \sqrt{3} )(2 - \sqrt{3} ) = 4 - 3 = 1$

${x}^{3} + {y}^{3} = (x + y) {}^{3} - 3xy(x + y) = (2 + \sqrt{3} + 2 - \sqrt{3} ) {}^{3} - 3 \times 1 \times (2 + \sqrt{3} + 2 - \sqrt{3} ) = {4}^{3} - 3 \times 4 = 64 - 12 = 52$

so,

$\frac{1}{ {x}^{3} } + \frac{1}{ {y}^{3} } = \frac{ {x}^{3} + {y}^{3} }{ {x}^{3} {y}^{3} } = 52 \: \: ans.$