## Introduction

A radical expression is any expression that contains a radical symbol. Radical symbols are use to find the square roots, cubic root, and higher.

To make simplifying radical expressions easier, learn the radical properties. Remember, numbers under the radical may not be less than zero because the squared, cubed, and higher roots of negative numbers are not real numbers.

Product Property of Radicals: $\sqrt{x} \times \sqrt{y}=\sqrt{xy}$

Quotient Property of Radicals: $\sqrt{\frac{x}{y}} =\frac{\sqrt{x}}{\sqrt{y}}$

To make radical properties easier to understand, substitute perfect squares for the variables in the properties.

\begin{align} \sqrt{16}\times \sqrt{81}&= \sqrt{16\times81}\\ 4 \times9&= \sqrt{196}\\ 36&=36 \end{align}

\begin{align} \sqrt{\frac{16}{4}} &=\frac{\sqrt{16}}{\sqrt{4}}\\ \sqrt{4}&=\frac{4}{2}\\ 2&=2 \end{align}

Radicals are not allowed in the denominator of fractions. To undo the radical, rationalize the denominator.

$\sqrt{\frac{x}{y}} =\frac{\sqrt{x}}{\sqrt{y}}=\frac{\sqrt{x}}{\sqrt{y}}\times\frac{\sqrt{y}}{\sqrt{y}} =\frac{\sqrt{xy}}{\sqrt{y^{2}}}=\frac{\sqrt{xy}}{y}$

When you rationalize the denominator, the radical cancels out.

$\frac{3}{\sqrt{5}} =\frac{3}{\sqrt{5}}\times\frac{\sqrt{5}}{\sqrt{5}} =\frac{3\sqrt{5}}{\sqrt{5^{2}}}=\frac{3\sqrt{5}}{5}$

Just like exponents, you can add, subtract, multiply, and divide expressions containing radicals to calculate solutions.

This expression is the sum of perfect squares. Simplify each perfect square then sum.

$\sqrt{36} + \sqrt{25} = 6 + 5 = 11$

If you can’t remove the radical, write the difference as a radical expression.

$\sqrt{64}-\sqrt{24}=8-\sqrt{4\times6}= 8-\sqrt{4}\times\sqrt{6}=8-2\sqrt{6}$

Remember to use the product rule when multiplying radical expressions.

For this problem, the product contains a radical. Use your calculator to determine the decimal answer, if needed.

$\sqrt{9}\times\sqrt{16}\times\sqrt{30}=3\times 4\times\sqrt{30}=12\times\sqrt{30}=12\sqrt{30}=65.73$

When simplifying expressions, apply the quotient rule.

This problem is the quotient of two perfect squares.

$\sqrt{\frac{9}{16}}=\frac{\sqrt{9}}{\sqrt{16}}=\frac{3}{4}$

## Distance Formula

To find the distance between two points on a graph, we use the distance formula.

Distance Formula

$d=\sqrt{\left( x_2-x_1\right) ^{2}+\left( y_2-y_1\right) ^{2}}$ Find the distance of the two points shown on the graph.

Use the distance formula to solve. Apply what you know about exponents and radicals to determine the answer.

\begin{align} d&=\sqrt{ \left(3 -2\right)^{2}+\left(6 -4\right)^{2}}\\ d&=\sqrt{ \left(1\right)^{2}+\left(2\right)^{2}}\\ d&=\sqrt{ 1 +4}\\ d&=\sqrt{5} \end{align}