Complex Algrebra

Question

If x=3+2√2 , find the value of x^3+1/x^3

Answer 1

Given that x = 3 + 2√2

We can multiply out the complex number to get the answer to x³ + 1/x³

x² = (3 + 2√2)(3 + 2√2) = 9 + 12√2 + 8 = 17 + 12√2

x³ = (17 + 12√2)(3 + 2√2) = 51 + 34√2 + 36√2 + 16 = 67 + 70√2

x³ = 67 + 70√2

Rationalizing the 1/x³ term...

1/x³ = 1/ ( 67 + 70√2 ) = (67 - 70√2) / (67² - 70² * 2) = (67 - 70√2) / (4489 - 9800 ) = (70√2 - 67)/5311

Therefore x³ + 1/x³ = 67 + 70√2 + (70√2 - 67)/5311