a)If z= a+b i find z^2 in the form X+ Y i
b)hence show that (z^2)* = (z*) ^2 for all complex numbers z
c) repeat a and b but for z^3 instead of z^2

Question

a)If z= a+b i find z^2 in the form X+ Y i 
b)hence show that (z^2)* = (z*) ^2 for all complex numbers z
c) repeat a and b but for z^3 instead of z^2

anonymous · Answer

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jamesj · Answer

So what do you have for part a)?

jamesj · Answer

$$ z^2 = (a + bi)^2 = a^2 + 2abi + b^2i^2 = ... what? $$

anonymous · Answer

^He's right.  Just distribute terms and recognize that since $i \equiv \sqrt{-1}$ then it follows that $i^2=-1$.

For part b, recall the definition of a complex conjugate: its' a reflection about the real axis.  Thus, just change every $i$ you see to a $-i$.  Generically, the number $c+di$ for constants $c$ and $d$ will have the complex conjugate $c-di$.  (Complex conjugates are denoted with a star (like you put) or a bar above).  Simplify both algebraically as you would normally to prove.

If you can get that far, try part c, and if you need help, just holler.