Help with complex numbers

10((1+i)/(1-i))^8 - 5((1-i)/(1+i))^6=15

 

Divide: (1+i) / (1-i) = i

Power: i ^ 8 = 1

Multiple: 10 * 1 = 10

Divide: (1-i) / (1+i) = -i

Power: (-i) ^ 6 = -1

Multiple: 5 * (-1) = -5

Subtract: 10 - (-5) = 15

This is sooooo long. I will divide it into 2 parts.

First part 

= \(10(\frac{1+i}{1-i})^8\)

=\(10(\frac{1^2+2i-1}{1-(-1)})^8 \)

= \(10(i)^8\)

=\(10\)

 

Second part

=\(5(\frac{1+i}{1-i})^6 \)

=\(5i^6\)

= \(-5 \)

 

First part - Second part 

= 10 - (-5)

=15