First Principles AI

Solution

\begin{aligned} & \boxed{\text{Write the complex number in } a+ib \text{ form:}} \\ \\ & z = \sqrt{2} - \sqrt{5}i \\ \\ & \boxed{\text{Formula for multiplicative inverse:}} \\ \\ & z^{-1} = \frac{1}{a+ib} = \frac{a-ib}{a^2+b^2} \\ \\ & \boxed{\text{Substitute } a=\sqrt{2} \text{ and } b=-\sqrt{5}:} \\ \\ & z^{-1} = \frac{\sqrt{2} - (-\sqrt{5})i}{(\sqrt{2})^2 + (-\sqrt{5})^2} \\ \\ & \boxed{\text{Simplify numerator:}} \\ \\ & = \frac{\sqrt{2} + \sqrt{5}i}{(\sqrt{2})^2 + (-\sqrt{5})^2} \\ \\ & \boxed{\text{Simplify denominator:}} \\ \\ & = \frac{\sqrt{2} + \sqrt{5}i}{2 + 5} = \frac{\sqrt{2} + \sqrt{5}i}{7} \\ \\ & \boxed{\text{Separate real and imaginary parts:}} \\ \\ & = \frac{\sqrt{2}}{7} + \frac{\sqrt{5}}{7}i \\ \\ & \boxed{\text{Write in ordered pair form:}} \\ \\ & \boxed{\left(\frac{\sqrt{2}}{7}, \frac{\sqrt{5}}{7}\right)} \end{aligned}