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

 Jul 8, 2024
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avatar+1275 
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Let's first focus on the first equation. 

Squaring both sides of the first equation, we find that

\(a^2 + 2ab + b^2 = 16 \)

 

Reaaranging the second equation from the problem, we get that

\(a^2 -ab+ b^2 = 6\)

 

Now, moving on, let's note something important. We have that

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

 

We already have all the terms needed to solve the problem. We know every single number. 

The first term parenthesis is the first equation, and the second parenthesis is the second equation. 

Thus, plugging in 6 and 4, we get that

\(a^3 + b^3 = 6*4 =24\)

 

So 24 is our answer. 

 

Thanks! :)

 Jul 8, 2024

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