Question #c5258

1 Answer
Jul 4, 2017

Use the fact that j^3 = -j.

Explanation:

Let j be the imaginary unit.
Since j^3 = -j,
we have
4^(5 + j^3) = 4^(5-j)
By the Product Rule for Exponents, this is equal to
4^5*4^(-j).
Now 4^5 = 1024, so we have
1024*4^(-j).

Now use the trig form: a^(jc) = cos(c*ln(a)) + jsin(c*ln(a)).
In this case, a = 4 and c = -1.
4^(-j) = cos(-ln(4)) + jsin(-ln(4))
Since cosine is EVEN and sine is ODD, we have...
4^(-j) = cos(ln(4)) - jsin(ln(4))

Don't forget to multiply by 1024 to obtain the final answer in the standard form. You should be able to easily identify the real and imaginary parts now.