Question

If a^2+1/a^2=79 where (a>0), find the value of a^3 +1/a^3

Answer 1

Note the binomial expansion,

(a + 1/a)³ = a ³ + 3a + 3/a + 1/a ³

so

a ³ + 1/a ³ = (a + 1/a)³ - 3 (a + 1/a)

Similarly,

(a + 1/a)² = a ² + 2 + 1/a ²

We're given a ² + 1/a ² = 79, so

(a + 1/a)² - 2 = 79

(a + 1/a)² = 81

a + 1/a = ±9

but a > 0, so we ignore the negative solution.

Then

a ³ + 1/a ³ = 9³ - 3×9 = 702