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Let $a$ and $b$ be complex numbers. If $a + b = 1$ and $a^2 + b^2 = 2,$ then what is $a^3 + b^3?$

 Jun 22, 2024
 #1
avatar+1230 
+1

First, let's take a look at the first equation we are given. 

\(a+b=1 \)

Squaring both sides, we have

\(a^2 + 2ab + b^2 = 1 \)

 

Now, we already know a^2 + b^2 is 2, so plugging that in, we have

\(2+2ab = 1\\ 2ab=-1\\ ab=-1/2\)

 

ab will come into handy later. 

 

Now, let's notcie something real quick. We have

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

 

We already know all the terms needed now! We have

\(a^3 + b^3 = (1) ( 2 +1/2)\\ a^3+b^3 = 5/2\)

 

So our answer is 5/2. 

 

Thanks! :)

 Jun 22, 2024

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