If a + 1/a=7, what is the value of a³ + 1/a³?
Recall that we can factor the sum of cubes:
x^3 + y^3 = (x+y)(x^2 - xy + y^2)
so a^3 + 1/a^3 = (a + 1/a)(a^2 - 1 + 1/a^2)
= 7(a^2 + 1/a^2 - 1)
= 7(a^2 + 1/a^2) - 7
now recall that x^2 + y^2 = (x+y)^2 - 2xy
so a^2 + 1/a^2 = (a + 1/a)^2 - 2(a)(1/a)
= 49 - 2
= 47
then:
7(a^2 + 1/a^2) - 7
= 7(47) - 7
= 322