Simplifying Radical Expressions

Objective: Rewrite a square root using a perfect-square factor.

Prerequisite skills: perfect squares and rational exponents. Factor the radicand as a perfect square times a remaining nonnegative factor. The square root of a product of nonnegative numbers is the product of their square roots. Extract the largest convenient square factor. Common error: taking half the radicand instead of its square root. Restrictions and interpretation: For real square roots the radicand must be nonnegative; √(x²) = |x|.

Objective: Add radical terms only after simplifying their radicands.

Radicals with the same remaining radicand act like like terms. Simplify each radical, then add or subtract their outside coefficients. Different radicands generally cannot be combined this way. Common error: adding radicands under one square root. Restrictions and interpretation: For real square roots the radicand must be nonnegative; √(x²) = |x|.

Objective: Use absolute value when extracting an even power.

A principal square root cannot be negative. Therefore √(x²) is |x|, not always x. A stated sign condition lets you replace the absolute value correctly. This matters when simplifying before evaluating. Common error: replacing √(x²) with x when x is negative. Restrictions and interpretation: For real square roots the radicand must be nonnegative; √(x²) = |x|. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.