Complex Inverse Ba1668

1. **State the problem:** Find the multiplicative inverse of the complex number $5 - 12i$. 2. **Recall the formula:** The multiplicative inverse of a complex number $z = a + bi$ is given by $$\frac{1}{z} = \frac{\overline{z}}{|z|^2}$$ where $\overline{z} = a - bi$ is the complex conjugate and $|z|^2 = a^2 + b^2$ is the modulus squared. 3. **Calculate the conjugate:** For $5 - 12i$, the conjugate is $5 + 12i$. 4. **Calculate the modulus squared:** $$|5 - 12i|^2 = 5^2 + (-12)^2 = 25 + 144 = 169$$ 5. **Write the inverse:** $$\frac{1}{5 - 12i} = \frac{5 + 12i}{169}$$ 6. **Simplify the fraction:** $$\frac{5}{169} + \frac{12}{169}i$$ **Final answer:** The multiplicative inverse of $5 - 12i$ is $$\frac{5}{169} + \frac{12}{169}i$$.