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

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 $(-a)\cdot (-a)={a}^{2}$.We usually will denote such dual answers as $\pm a$.

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.

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

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.

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

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.

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

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

**Example 5:**

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

Simplify both radicals:

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

Now, the radicands are the same.

So, we can add using the distributive property.