Can someone help me solve this problem? ((3 + i) ^(2) )/((1 + 2i)^(2) ) How do I begin to solve this?

Question

Can someone help me solve this problem?


`((3 + i) ^(2) )/((1 + 2i)^(2) )`
How do I begin to solve this?

Answer 1

(3 + i)^2 * (1 - 2i)^2
==============
( 1 + 2i)^2 * ( 1 - 2i)^2

Let's work on the denominator first.
((1 + 2i)(1 - 2i))^2
(1 - 4i^2)^2
(1 + 4)^2
25

now work on the numerator which does not look very nice.

((3 + i)(1 - 2i))^2
(3 - 6i + i - 2(i)^2 )^2
(3 - 6i + i + 2)^2
(5 - 5i)^2
25(1 - i)^2
25(1 - 2i + i^2)
25 (-2i)
-50i

Now put the two parts together
-50i / 25 = - 2i

There might be a shorter way to do this, but I don't know what it might be.