Chapter 1.6, Problem 63E

Solution

To write: The expression $\frac{a+bi}{c-di}$ as a complex number in standard form.

  $\frac{ac-bd}{c^2+d^2}+\frac{ad+bc}{c^2+d^2}i$

Given information:

The expression, $\frac{a+bi}{c-di}$ .

Concept used:

Distributive law: $a\left(b+c\right)=ab+ac$

Calculation:

Multiply the numerator and denominator by the conjugate of $c-di$ , and simplify the resulting expression.

  $\begin{array}{c}\frac{a+bi}{c-di}=\frac{\left(a+bi\right)\left(c+di\right)}{\left(c-di\right)\left(c+di\right)}\\ =\frac{ac-bd+\left(ad+bc\right)i}{c^2+d^2}\\ =\frac{ac-bd}{c^2+d^2}+\frac{ad+bc}{c^2+d^2}i\end{array}$

Conclusion:

The given expression in standard form of complex number is $\frac{ac-bd}{c^2+d^2}+\frac{ad+bc}{c^2+d^2}i$ .