Simplifying Radicals

A radical expression is considered fully simplified when the number inside the root (the radicand) has no perfect square factors (for square roots) or perfect cube factors (for cube roots), and there are no radicals in the denominator of a fraction.

Factoring Out Perfect Squares

To simplify a square root, look for the largest perfect square ($1, 4, 9, 16, 25, 36, \dots$) that divides evenly into the radicand.

Example: Simplify $\sqrt{72}$

1. Find the largest perfect square factor of $72$. Here, $72 = 36 \times 2$, and $36$ is a perfect square.
2. Rewrite the radical using this multiplication: $\sqrt{72} = \sqrt{36 \times 2}$
3. Separate the roots and simplify the perfect square: $\sqrt{36} \times \sqrt{2} = 6\sqrt{2}$

So, $\sqrt{72}$ simplified is $6\sqrt{2}$.

(Note: The same process applies to cube roots, but you look for perfect cube factors like $8, 27, 64$, etc.)

Rationalizing the Denominator

In mathematics, it is standard practice to write fractions without radicals in the denominator. The process of removing the radical from the bottom of a fraction is called rationalizing the denominator.

To do this, multiply both the numerator and the denominator by a radical that will eliminate the root on the bottom.

Example: Rationalize $\frac{5}{\sqrt{3}}$

1. Multiply the top and bottom by $\sqrt{3}$ to make the denominator a perfect square: $\frac{5}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}$
2. Multiply across: $\frac{5\sqrt{3}}{\sqrt{9}}$
3. Simplify the denominator: $\frac{5\sqrt{3}}{3}$

The expression is now fully simplified because there is no longer a square root in the denominator.