Explanation:
(a + 1/a)³ = a³ + 3a + 3/a + 1/a³
We are given that a + 1/a = 4, so we can substitute that into the identity and simplify:
(a + 1/a)³ = (4)³ = 64
a³ + 3a + 3/a + 1/a³ = 64
a³ + 1/a³ + 3(a + 1/a) = 64
a³ + 1/a³ + 3(4) = 64
a³ + 1/a³ = 52
Therefore, we have proved that a³ + 1/a³ = 52 when a + 1/a = 4.