Expert: Paul Klarreich - 5/6/2009

Question
Hi, i was just wondering if you could belp me with simplifying (-1+i)^16 using euler form.

I understand how to convert -1+i to euler form but i am unsure about how to deal with the exponent.

Any help would be greatly appreciated, thanks

Answer
Questioner:   Olivia
Country:  Australia
Category:  Advanced Math
Private:  No
 
Subject:  simplifying complex numbers
Question:  Hi, i was just wondering if you could belp me with simplifying (-1+i)^16 using euler form.

I understand how to convert -1+i to euler form but i am unsure about how to deal with the exponent.

Any help would be greatly appreciated, thanks
.......................................
Hi, Olivia,

I assume you have converted -1 + i  to  sqrt(2) exp(pi/4)

Now the nice thing about this exponent is that it behaves as if it were an exponent. (Duhhhh, as we say in the US.  I am sure you have a similar expression.)

That means all the exponent laws apply, including the one that says:

(x^a)^b = x^(ab) -- multiply exponents.

Now:

(-1 + i)^16 =
(sqrt(2) exp(pi/4))^16 =
 
(2^(1/2) exp(pi/4))^16 =

2^8 exp(16pi/4) =

512 exp(4pi) =

Now you will reconvert:  exp(4pi) = exp(0) = 1 (plus 0i, but...)

= 512(1) = 512

That's it.