Question

I need help with this practice problem solving from my online ACT prep guide

Answer 1

Case: Complex number simplification

Given:

`(4+i)/(-2+3i)`

Required: To simplify the sum to a standard form for complex numbers

Method: Simplifaction by rationalization.

The rationalization process includes multiplying the top and bottom by the conjugate of the denominator i.e changing the separating operation to the opposite.

In this case the conjugate will be: (-2 - 3i)

`\begin{gathered} (4+i)/(-2+3i)*((-2-3i))/((-2-3i)) \\ ((4+i)(-2-3i))/((-2+3i)(-2-3i)) \\ (4(-2-3i)+i(-2-3i))/(-2(-2-3i)+3i(-2-3i)) \\ (-8-12i-2i-3i^2)/(4+6i-6i-9i^2) \\ (-8-14i-3i^2)/(4-9i^2) \\ \text{But i}^2=-1 \\ \frac{-8-14i-3(-1)^{}}{4-9(-1)^{}} \\ \frac{-8-14i+3^{}}{4+9^{}} \\ (-5-14i)/(13) \end{gathered}`

Final answer:

`\begin{gathered} (-5-14i)/(13)\text{ OR} \\ -(5)/(13)-(14i)/(13) \end{gathered}`