Question

\frac{(3 - 2i)(2 + 3i)}{(1 + i \sqrt{2)}(2 - i) } \ \textless \ br /\ \textgreater \ 
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Answer 1

A simple property -

z. (z conjugate) =|z|2

So, (2–3i)*(2+3i)=(√(22+32))2=13

Similarly, (3+4i)*(3–4i)=(√(32+42))2=25

Therefore, given expression =13/25

Which is clearly purely real.

Proof of above property:

Let z= x+iy

then z conjugate= x-iy

z*(z conjugate) = x2−i2y2+2ixy−2ixy

=x2+y2=|z|2

Note: This property is very useful while playing with complex numbers.