Let $a$ and $b$ be complex numbers. If $a + b = 4$ and $a^2 + b^2 = 6,$ then what is $a^3 + b^3?$

onyuIee Jul 24, 2024

#1**+1 **

Let's use a whole number of equations to solve this problem.

First off, we know that \((a+b)^2=a^2+2ab+b^2\)

We ALSO know that \((a+b)^2=4^2=16\)

Thus, we have the equation \(a^2+2ab+b^2=16\)

We are given a^2+b^2 in the problem, so plugging that in, we get

\(6+2ab=16\)

We can now find ab. This will come in handy later.

\(2ab=10\\ ab=5\)

Alright, let's move on to what we are TRYING to find.

Let's note that we can split a^3+b^3 into

\(a^3+b^3=\left(a+b\right)\left(a^2-ab+b^2\right)\)

WAIT! We already know the values of every number! Plugging in 4, 6, and 5, we get

We have

\((4)(6-5)=4\)

So 4 SHOULD be the answer.

I might have a mistake during the calculations...not sure.

Thanks! :)

NotThatSmart Jul 25, 2024