Understanding the Square Root of the Absolute Value of a Complex Number

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: \( \sqrt[4]{a^2 + b^2} \)