The imaginary unit and cyclic powers

Build the foundation for complex-number algebra through the imaginary unit and its repeating powers. This focused practice set explains how real numbers sit inside the complex number system, why a number whose square is negative is not real and how negative real radicands can be written using the imaginary unit.

The questions distinguish a single principal square-root value from the two solutions of a quadratic equation. They also develop the four-step power cycle, extend it to negative integer exponents and use cancellation across a complete cycle. Error-analysis tasks address two common mistakes: multiplying principal square roots of negative numbers as if the nonnegative real radical rule applied unchanged, and replacing the square root of a squared parameter by the parameter without checking its sign.

Prerequisites are integer exponent rules, radicals and elementary quadratic equations. No polar form, trigonometry or general complex powers are used. Both single-select and multiselect questions progress from definitions to symbolic reasoning with a stated real parameter. Explanations show the short calculation behind each answer and make conventions explicit. Work through the cycle or square your proposed root as a check rather than memorizing isolated large powers. General complex arithmetic and quadratic factorization are developed in the following sets.