Method guideAlgebra IIntermediate9 min read

Simplifying radicals by pulling out perfect powers

Factor the radicand into a perfect power times what remains. Pull only complete groups through the radical bar.

\sqrt{ab}=\sqrt a\sqrt b\quad(a,b\ge0)

Method guide

How to recognize the method, run it, and know when it is the wrong choice.

Updated July 13, 2026

MethodStart here

For square roots, every pair of equal factors contributes one factor outside.

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.

\sqrt{72}=\sqrt{36\cdot2}=6\sqrt2

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.

\sqrt{x^2}=|x|

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.

3\sqrt5-\sqrt5=2\sqrt5

Worked exampleExtract complete pairs

200x^5=100x^4\cdot2x

Separate the largest perfect-square factor.

\sqrt{200x^5}=\sqrt{100x^4}\sqrt{2x}

Use the product property.

\sqrt{100x^4}=10x^2

Take the principal square root under the nonnegative assumption.

10x^2\sqrt{2x}

Keep the unpaired factors inside.

Result

\boxed{10x^2\sqrt{2x}}

Watch for

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