Lesson 1: Properties of Radicals

Property

A radical expression is considered simplified if there are

• no factors in the radicand have perfect powers of the index
• no fractions in the radicand
• no radicals in the denominator of a fraction

Examples

• Is $\sqrt{50}$ simplified? No, because $\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}$. It contained a perfect square factor.
• Is $\sqrt{\dfrac{5}{9}}$ simplified? No, because it has a fraction in the radicand. It simplifies to $\dfrac{\sqrt{5}}{\sqrt{9}} = \dfrac{\sqrt{5}}{3}$.
• Is $\dfrac{7}{\sqrt{3}}$ simplified? No, because there is a radical in the denominator. It must be rationalized to become $\dfrac{7\sqrt{3}}{3}$.

Explanation

A radical is fully simplified when it's completely tidy. This means no perfect squares (or cubes, etc.) are left inside, no fractions are under the radical sign, and no radicals are hiding in the denominator of a fraction.

Property

If $a, b$ are non-negative real numbers, then $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$.

Simplify a square root using the product property.

1. Find the largest perfect square factor of the radicand. Rewrite the radicand as a product using the perfect-square factor.
2. Use the product rule to rewrite the radical as the product of two radicals.
3. Simplify the square root of the perfect square.

Examples

• To simplify $\sqrt{48}$: Find the largest perfect square factor, which is 16. Rewrite as $\sqrt{16 \cdot 3}$, which becomes $\sqrt{16} \cdot \sqrt{3}$, simplifying to $4\sqrt{3}$.