- #1
chwala
Gold Member
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- Homework Statement
- If ##z=a+bi##, where ##a## and ##b## are real, use binomial theorem to find the real and imaginary parts of ##z^5##
- Relevant Equations
- Complex numbers
This is also pretty easy,
##z^5=(a+bi)^5##
##(a+bi)^5= a^5+\dfrac {5a^4bi}{1!}+\dfrac {20a^3(bi)^2}{2!}+\dfrac {60a^2(bi)^3}{3!}+\dfrac {120a(bi)^4}{4!}+\dfrac {120(bi)^5}{5!}##
##(a+bi)^5=a^5+5a^4bi-10a^3b^2-10a^2b^3i+5ab^4+b^5i##
##\bigl(\Re (z))=a^5-10a^3b^2+5ab^4##
##\bigl(\Im (z))= 5a^4b-10a^2b^3+b^5##
Any other variation, combinations may also work...
##z^5=(a+bi)^5##
##(a+bi)^5= a^5+\dfrac {5a^4bi}{1!}+\dfrac {20a^3(bi)^2}{2!}+\dfrac {60a^2(bi)^3}{3!}+\dfrac {120a(bi)^4}{4!}+\dfrac {120(bi)^5}{5!}##
##(a+bi)^5=a^5+5a^4bi-10a^3b^2-10a^2b^3i+5ab^4+b^5i##
##\bigl(\Re (z))=a^5-10a^3b^2+5ab^4##
##\bigl(\Im (z))= 5a^4b-10a^2b^3+b^5##
Any other variation, combinations may also work...
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