Simplifying radicals by pulling out perfect powers

Look for the largest useful perfect square

A radicand can be split into a perfect-square factor and a leftover factor. Using the largest square usually reaches simplest form in one move.

Prime factorization is slower but dependable when the square factor is not obvious.

Variables need domain awareness

For real variables, √(x²) equals |x|, not automatically x, because the principal square root is nonnegative.

Many introductory exercises quietly assume variables are nonnegative. State that assumption or keep the absolute value when it matters.

Only like radicals combine

After simplification, radicals combine like algebraic terms when their indices and radicands match. Coefficients add; radicands do not.

Do not split a sum inside a radical: √(a + b) is generally not √a + √b.

Worked example

Common mistakes

• Pulling out a factor that is not a complete square.
• Writing √(x²) = x without a domain assumption.
• Adding unlike radicands.

Keep these ideas

• Extract perfect powers.
• Principal roots are nonnegative.
• Combine only matching radicals.