# Reduced Radical Form

An expression that contains a radical sign ( $\sqrt{}$ ) is said to be in reduced radical form if the radicand–that's the number under the radical sign–does not contain any perfect squares (or perfect cubes, if it's in the cube root sign.)

You can use the following property to simplify a square root.

Product Property of Square Roots

For all real numbers $a$ and $b$ , $\sqrt{ab}=\sqrt{a}\cdot \sqrt{b}$ .

That is, the square root of the product is the same as the product of the square roots.

Example 1:

Simplify.

$\sqrt{18}$

Factor the radicand using perfect squares.

We know that $9×2=18$ . So, rewrite $18$ as the product of $9$ and $2$ .

$\sqrt{18}=\sqrt{9\cdot 2}$

Now use the product property of square roots.

$=\sqrt{9}\cdot \sqrt{2}$

Simplify.

$=3\sqrt{2}$

Example 2:

Simplify.

$\sqrt{252}$

Factor the radicand using perfect squares.

We know that $36×7=252$ . So, rewrite $252$ as the product of $36$ and $7$ .

$\sqrt{252}=\sqrt{36\cdot 7}$

Now use the product property of square roots.

$=\sqrt{36}\cdot \sqrt{7}$

Simplify.

$\begin{array}{l}=6\cdot \sqrt{7}\\ =6\sqrt{7}\end{array}$