First Principles AI

Solution

\begin{aligned} & \boxed{\text{Write the complex number in } a+ib \text{ form:}} \\ \\ & z = -4 + 7i \\ \\ & \boxed{\text{Formula for multiplicative inverse:}} \\ \\ & z^{-1} = \frac{1}{a+ib} = \frac{a-ib}{a^2+b^2} \\ \\ & \boxed{\text{Substitute } a=-4 \text{ and } b=7:} \\ \\ & z^{-1} = \frac{-4 - 7i}{(-4)^2 + 7^2} \\ \\ & \boxed{\text{Simplify denominator:}} \\ \\ & = \frac{-4 - 7i}{16 + 49} = \frac{-4 - 7i}{65} \\ \\ & \boxed{\text{Separate real and imaginary parts:}} \\ \\ & = -\frac{4}{65} - \frac{7}{65}i \\ \\ & \boxed{\text{Write in ordered pair form:}} \\ \\ & \boxed{\left(-\frac{4}{65}, -\frac{7}{65}\right)} \end{aligned}