7-4 Simplifying Radical Expressions

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

For any integer $n \geq 2$,

• when the index $n$ is odd, $\sqrt[n]{a^n} = a$
• when the index $n$ is even, $\sqrt[n]{a^n} = |a|$

We must use the absolute value signs when we take an even root of an expression with a variable in the radical.

Examples

• Simplify $\sqrt{y^2}$. Since the index (2) is even, we must use an absolute value. The result is $|y|$.

Property

For any positive real numbers $a$ and $b$, and positive integer $n \geq 2$:

Product Property: $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}$