How to simplify $\displaystyle \frac {3+4i}{5-2i} $? Thanks.
Hello,
A conjugate of a complex number $\displaystyle a+ib$ is defined to be $\displaystyle a-ib$
What's good about it is if you multiply a complex number by its conjugate :
$\displaystyle (a+ib)(a-ib)=a^2+b^2$ (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 :
$\displaystyle \frac{3+4i}{5-2i}=\frac{3+4i}{5-2i} \cdot \frac{5+2i}{5+2i}=\frac{(3+4i)(5+2i)}{5^2+2^2}$
expand the numerator and separate real and imaginary parts.