Question

Question #c5258

Answer 1

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.