Calculating √|z| for a Complex Number z = a + bi: A Step-by-Step Guide with Commentary
\( \sqrt{|z|} \) | = | \( \sqrt{|a + bi|} \) | The square root of the absolute value of ‘z’. |
= | \( \sqrt{\sqrt{a^2 + b^2}} \) | Expressing |z| as the square root of a² + b². | |
= | \( (a^2 + b^2)^{\frac{1}{2} \cdot \frac{1}{2}} \) | Squaring the square root to simplify. | |
= | \( (a^2 + b^2)^{\frac{1}{4}} \) | Simplified to the fourth root of a² + b². | |
= | \( \sqrt[4]{a^2 + b^2} \) | The final simplified form of the square root of |z|. |
This guide provides a detailed walkthrough of the process to find the square root of the absolute value of a complex number, with commentary that makes each step clear and comprehensible.
Final Answer: \( \sqrt[4]{a^2 + b^2} \)