Radical Rules

High School

Definition

Rules that tell you how to carry out operations on radicals. Keep in mind that radicals can be rewritten with exponents, so some of the rules you'll encounter will move the radicals into a number's exponent.

Worked examples

\(\sqrt{16} \cdot \sqrt{9} = \sqrt{16 \cdot 9} = \sqrt{144} = 12\)

Multiply radicals with the same index by multiplying what's inside, then simplify.

\(\frac{\sqrt{50}}{\sqrt{2}} = \sqrt{\frac{50}{2}} = \sqrt{25} = 5\)

Divide radicals with the same index by dividing what's inside, then simplify.

\(\sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2}\)

Factor out perfect squares to simplify a radical into its simplest form.

Common mistakes

\(\sqrt{a + b} = \sqrt{a} + \sqrt{b}\) → \(\sqrt{a + b}\) cannot be split You can only split multiplication or division under a radical, never addition or subtraction.
\(\sqrt{3} \cdot \sqrt[3]{3} = \sqrt{9}\) → \(\sqrt{3} \cdot \sqrt[3]{3} = 3^{1/2} \cdot 3^{1/3} = 3^{5/6}\) Radicals with different indices cannot be multiplied directly; convert to exponents first.
\(\sqrt{x^2} = x\) → \(\sqrt{x^2} = |x|\) The square root returns the non-negative result, so you need absolute value when x might be negative.

Where you'll use it next

Radical rules are essential for simplifying expressions in algebra, solving radical equations, working with rational exponents, and handling square roots in the quadratic formula and distance formula in geometry.

Found in 1 StudyPug lesson

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