# Thread: Complex simpification

1. ## Complex simpification

How to simplify $\frac {3+4i}{5-2i}$? Thanks.

2. Originally Posted by tttcomrader
How to simplify $\frac {3+4i}{5-2i}$? Thanks.
Multiply both the numerator and the denominator by the complex conjugate of the denominator. ie:

$\frac{3+4i}{5-2i} \times \frac{5+2i}{5+2i}$

3. Hello,
Originally Posted by tttcomrader
How to simplify $\frac {3+4i}{5-2i}$? Thanks.
A conjugate of a complex number $a+ib$ is defined to be $a-ib$
What's good about it is if you multiply a complex number by its conjugate :
$(a+ib)(a-ib)=a^2+b^2$ (using the identity (x-y)(x+y)=x²-y²) and that's a real number.

So the main idea is to multiply both numerator and denominator by the conjugate of the denominator :

$\frac{3+4i}{5-2i}=\frac{3+4i}{5-2i} \cdot \frac{5+2i}{5+2i}=\frac{(3+4i)(5+2i)}{5^2+2^2}$
expand the numerator and separate real and imaginary parts.