write (1+i)^13 in a+bi form

Question

write (1+i)^13 in a+bi form where b and a are real numbers

Answer 1

I am assuming that you have learned the polar form of complex numbers. This form, and a theorem, makes it easy to raise complex numbers to any power.

 

your number in polar for is √2(cos 45 + i sin 45).

 

To raise this to the 13th power you get

 

√2¹³*(cos 13*45 + i sin 13*45)

 

√2¹³ = 2⁶√2 = 64√2.

cos 585 + i sin 585 = cos 225+i sin 225 using coterminal angles = -√2 / 2 - i √2 / 2.

 

Multiplying these you get -64-64 i

 

Hope this helped.