Before you can simplify a radical expression, you have to know the important properties of radicals .

## PRODUCT PROPERTY OF SQUARE ROOTS

For all real numbers $a$ and $b$ ,

$\sqrt{a}\cdot \sqrt{b}=\sqrt{a\cdot b}$

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

There's an analogous quotient property:

For all real numbers $a$ and $b$ , $b\ne 0$ :

$\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}$

Since a negative number times a negative number is always a positive number, you need to remember when taking a square root that the answer will be both a positive and a negative number or expression. For example $a\cdot a={a}^{2}$ , and also $\left(-a\right)\cdot \left(-a\right)={a}^{2}$ .We usually will denote such dual answers as $±a$ .

The idea here is to find a perfect square factor of the radicand, write the radicand as a product, and then use the product property to simplify.

Example 1:

Simplify. $\sqrt{45}$

$9$ is a perfect square, which is also a factor of $45$ .

$\sqrt{45}=\sqrt{9\cdot 5}$

Use the product property.

$\begin{array}{l}\sqrt{9\cdot 5}=\sqrt{9}\cdot \sqrt{5}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=±3\sqrt{5}\end{array}$

If the number under the radical has no perfect square factors, then it cannot be simplified further.  For instance the number  $\sqrt{17}$ cannot be simplified further because the only factors of $17$ or $17$ and $1$ .  So, there are no perfect square factors other than $1$ .

Example 2:

Simplify. $\frac{\sqrt{12}}{\sqrt{3}}$

Use the quotient property to write under a single square root sign.

$\frac{\sqrt{12}}{\sqrt{3}}=\sqrt{\frac{12}{3}}$

Divide.

$\begin{array}{l}=\sqrt{4}\\ =±2\end{array}$

An expression is considered simplified only if there is no radical sign in the denominator. If we do have a radical sign, we have to rationalize the denominator . This is achieved by multiplying both the numerator and denominator by the radical in the denominator. Note that here, we're just multiplying by a special form of $1$ , so it doesn't change the value of the expression.

Example 3:

Simplify. $\frac{\sqrt{5}}{\sqrt{6}}$

$\frac{\sqrt{5}}{\sqrt{6}}=\frac{\sqrt{5}}{\sqrt{6}}\cdot \frac{\sqrt{6}}{\sqrt{6}}$

Simplify.

$=\frac{\sqrt{30}}{6}$

Sometimes we need to use a combination of steps.

Example 4:

Simplify. $\sqrt{\frac{21}{9}}$

$21$ and $9$ have a common factor of $3$ , so reduce the fraction under the radical.

$\sqrt{\frac{21}{9}}=\sqrt{\frac{7}{3}}=\frac{\sqrt{7}}{\sqrt{3}}$

Now rationalize the denominator.

$\frac{\sqrt{7}}{\sqrt{3}}\cdot \frac{\sqrt{3}}{\sqrt{3}}=\frac{\sqrt{21}}{3}$

We can only add or subtract two radical expressions if the radicands are the same.  For example, $\sqrt{17}+\sqrt{13}$ cannot be simplified any further. But we can simplify $5\sqrt{2}+3\sqrt{2}$ by using the distributive property , because the radicands are the same.

$5\sqrt{2}+3\sqrt{2}=\left(5+3\right)\sqrt{2}=8\sqrt{2}$

Be careful!  Sometimes, the radicands look different, but it's possible to simplify and get the same radicand.

Example 5:

Simplify. $\sqrt{50}+\sqrt{32}$

$\begin{array}{l}\sqrt{50}+\sqrt{32}=\sqrt{25\cdot 2}+\sqrt{16\cdot 2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=±5\sqrt{2}±4\sqrt{2}\end{array}$