Radicals and Rational Expressions

Simplifying Radicals

Simplify √72 by factoring out the largest perfect square. Write 72 = 36·2, so √72 = √(36·2) = 6√2. Verify by squaring: (6√2)² = 36·2 = 72.

Example

Factor out the largest perfect square and keep the radical exact.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

72\sqrt{72}

Step 2 — Factor radicand

Factor the radicand: 72 = 36 x 2.

72=36⋅2\sqrt{72} = \sqrt{ \hl{36} \cdot \hl{2} }

Step 3 — Extract square root

Take out the square root: square root of 36 is 6.

36=6\sqrt{36} = \hl{6}

Step 4 — Simplified radical

Write the exact result: 6 times square root of 2.

62\hl{6} \hlmath{\sqrt{2}}
simplify-radical Find the largest perfect-square factor of the radicand. Factor it out, take its square root, and multiply.