First Principles AI

Solution

\begin{aligned} & \boxed{\text{First expand } (4+3i)^2:} \\ \\ & (4+3i)^2 = 4^2 + 2(4)(3i) + (3i)^2 \\ & = 16 + 24i + 9i^2 = 16 + 24i - 9 = 7 + 24i \\ \\ & \boxed{\text{Now divide by } (4-3i) \text{ using conjugate } (4+3i):} \\ \\ & \frac{7+24i}{4-3i} \cdot \frac{4+3i}{4+3i} = \frac{(7+24i)(4+3i)}{(4-3i)(4+3i)} \\ \\ & \boxed{\text{Expand numerator:}} \\ \\ & (7+24i)(4+3i)=28+21i+96i+72i^2 \\ & = 28 + 117i - 72 = -44 + 117i \\ \\ & \boxed{\text{Simplify denominator:}} \\ \\ & (4-3i)(4+3i)=4^2+3^2=16+9=25 \\ \\ & \boxed{\text{Write in } a+ib \text{ form:}} \\ \\ & \frac{-44+117i}{25} = -\frac{44}{25} + \frac{117}{25}i \end{aligned}