Solution of the Week #563 - Powers

a+b = s

a*b = p

What is a^4 + b^4 in terms of s and p?

Let’s start by squaring (a+b):

s^2 = a^2 + 2ab + b^2

Let’s isolate (a^2+b^2)

(a^2+b^2) = s^2 – 2p

Let’s square both sides:

a^4 + 2a^2b^2 + b^4 = s^4 – 4ps^2 + 4p^2

Again let’s isolate (a^4+b^4)

a^4 + b^4 = s^4 – 4ps^2 + 4p^2 – 2p^2

a^4 + b^4 = s^4 – 4ps^2 + 2p^2

And of course this works, and is real, even if the choice of s and p mean that a and b must be complex. For instance if the sum and product were 1 and 2 respectively, the sum of the fourth powers would be, using the formula, 1. But if you calculate a and b you would find messy expressions involving i and the square root of 7.