Question

find the value of x cube + 1 by x cube

Answer 1

given the value of x = 2 + √3

therefore (x)³ = (2 + √3)³

by using identity (a + b)³ = a³ + 3a²b + 3ab² + b³

= (2)³ + 3(2)²(√3) + 3(2)(√3)² + (√3)³

= 8 + 12√3 + 6(3) + 3√3

= 8 + 18 + 12√3 + 3√3

= 26 + 15√3

➡ 1/x³ = 1/(26 + 15√3)

$\tt = \frac{1}{26 + 15 \sqrt{3} } \\ \\ \tt = \frac{1}{26 + 15 \sqrt{3} } \times \frac{26 - 15 \sqrt{3} }{26 - 15 \sqrt{3} } \\ \\ \tt = \frac{1(26 - 15 \sqrt{3}) }{(26 + 15 \sqrt{3} )(26 - 15 \sqrt{3} )} \\ \\ \tt = \frac{26 - 15 \sqrt{3} }{( {26})^{2} - ( {15 \sqrt{3} })^{2} } \\ \\ \tt = \frac{26 - 15 \sqrt{3} }{676 - 675} \\ \\ \tt = 26 - 15 \sqrt{3}$

➡ x³ + 1/x³ = (26 + 15√3) + (26 - 15√3)

= 26 + 15√3 + 26 - 15√3

= 52