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