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Simplifying Radical Expressions

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 $b$ , $b \neq 0$ :

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

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 $\pm a$ .

SIMPLIFYING RADICALS

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.

$\begin{array}{l} \sqrt{9 \cdot 5} = \sqrt{9} \cdot \sqrt{5} \\ = \pm 3 \sqrt{5} \end{array}$

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.

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

Now rationalize the denominator.

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

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:

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

Now, the radicands are the same.

So, we can add using the distributive property.