Question 420878
The easy way to raise a complex number to a power is DeMoivre's Theorem:
For any complex number
{{{z = r(cos(x) + i*sin(x))}}}
then
{{{z^n = r^n*(cos(n*x) + i*sin(n*x))}}}<br>
For your complex number, {{{cos(45)+i*sin(45)}}}, there is no visible r. So the r must be a 1. Writing your expression with a visible r we get:
{{{(1(cos(45) + i*sin(45)))^8}}}
Now we can use DeMoivre's Theorem to rewrite your expression:
1^8(cos(8*45) + i*sin(8*45))
which simplifies as follows:
1(cos(360) + i*sin(360))
1(1 + i*0)
1(1)
1
So (cos45+isin45)^8 = 1