I'm doing online homework, and the question is

Find all solutions of the equation. Check your solutions in the original equation. (Enter your answers as a comma-separated list.)

x3 + 512 = 0

The correct answers, verified by the system, are:
\({-8, 4-4\sqrt{3}i, 4+4\sqrt{3}i}\)

I don't understand how the last 2 are correct, because i X i = -1, so \((4+4\sqrt{3}i ) * (4+4\sqrt{3}i )\) should be a real number, but \((4+4\sqrt{3}i ) * (4+4\sqrt{3}i ) * (4+4\sqrt{3}i )\)should be an imaginary number, correct?

 

Wrong. Multiply it out and see how the terms with the imaginary unit cancel each other out.

 

DOH! Yep, they sure do.
Thanks

EDIT:
I was also wrong about \((4+4\sqrt{3}i ) * (4+4\sqrt{3}i )\) being a real number.

 

No sweat. We all have occasional lapses where our intuition plays tricks on us.

 

Just want to chime and say that it's good to see someone asking "does the answer make sense?", and that includes asking it of the "book's" answers, since the book will be wrong from ime to time.