### Learning Outcomes

- Simplify radical expressions using factoring
- Simplify radical expressions using rational exponents and the laws of exponents
- Define [latex]\sqrt{x^2}=|x|[/latex] and apply it when simplifying radical expressions

**Radical expressions** are expressions that contain radicals. Radical expressions come in many forms, from simple and familiar, such as[latex] \sqrt{16}[/latex], to quite complicated, as in [latex] \sqrt[3]{250{{x}^{4}}y}[/latex].

To simplify complicated radical expressions, we can use some definitions and rules from simplifying exponents. Recall the **Product Raised to a Power Rule** from when you studied exponents. This rule states that the product of two or more non-zero numbers raised to a power is equal to the product of each number raised to the same power. In math terms, it is written [latex]\left(ab\right)^{x}=a^{x}\cdot{b}^{x}[/latex]. So, for example, you can use the rule to rewrite [latex] {{\left( 3x \right)}^{2}}[/latex] as [latex] {{3}^{2}}\cdot {{x}^{2}}=9\cdot {{x}^{2}}=9{{x}^{2}}[/latex].

Now instead of using the exponent [latex]2[/latex], use the exponent [latex] \frac{1}{2}[/latex]. The exponent is distributed in the same way.

[latex] {{\left( 3x \right)}^{\frac{1}{2}}}={{3}^{\frac{1}{2}}}\cdot {{x}^{\frac{1}{2}}}[/latex]

And since you know that raising a number to the [latex] \frac{1}{2}[/latex] power is the same as taking the square root of that number, you can also write it this way.

[latex] \sqrt{3x}=\sqrt{3}\cdot \sqrt{x}[/latex]

Look at that—you can think of any number underneath a radical as the *product of separate factors*, each underneath its own radical.

### A Product Raised to a Power Rule or sometimes called The Square Root of a Product Rule

For any real numbers *a* and *b*, [latex] \sqrt{ab}=\sqrt{a}\cdot \sqrt{b}[/latex].

For example: [latex] \sqrt{100}=\sqrt{10}\cdot \sqrt{10}[/latex], and [latex] \sqrt{75}=\sqrt{25}\cdot \sqrt{3}[/latex]

This rule is important because it helps you think of one radical as the product of multiple radicals. If you can identify perfect squares within a radical, as with [latex] \sqrt{(2\cdot 2)(2\cdot 2)(3\cdot 3})[/latex], you can rewrite the expression as the product of multiple perfect squares: [latex] \sqrt{{{2}^{2}}}\cdot \sqrt{{{2}^{2}}}\cdot \sqrt{{{3}^{2}}}[/latex].

The square root of a product rule will help us simplify roots that are not perfect as is shown the following example.

### Example

Simplify. [latex] \sqrt{63}[/latex]

The final answer [latex] 3\sqrt{7}[/latex] may look a bit odd, but it is in simplified form. You can read this as “three radical seven” or “three times the square root of seven.”

The following video shows more examples of how to simplify square roots that do not have perfect square radicands.

Before we move on to simplifying more complex radicals with variables, we need to learn about an important behavior of square roots with variables in the radicand.

Consider the expression [latex] \sqrt{{{x}^{2}}}[/latex]. This looks like it should be equal to *x*, right? Test some values for *x* and see what happens.

In the chart below, look along each row and determine whether the value of *x* is the same as the value of [latex] \sqrt{{{x}^{2}}}[/latex]. Where are they equal? Where are they not equal?

After doing that for each row, look again and determine whether the value of [latex] \sqrt{{{x}^{2}}}[/latex] is the same as the value of [latex]\left|x\right|[/latex].

[latex]x[/latex] | [latex]x^{2}[/latex] | [latex]\sqrt{x^{2}}[/latex] | [latex]\left|x\right|[/latex] |
---|---|---|---|

[latex]−5[/latex] | [latex]25[/latex] | [latex]5[/latex] | [latex]5[/latex] |

[latex]−2[/latex] | [latex]4[/latex] | [latex]2[/latex] | [latex]2[/latex] |

[latex]0[/latex] | [latex]0[/latex] | [latex]0[/latex] | [latex]0[/latex] |

[latex]6[/latex] | [latex]36[/latex] | [latex]6[/latex] | [latex]6[/latex] |

[latex]10[/latex] | [latex]100[/latex] | [latex]10[/latex] | [latex]10[/latex] |

Notice—in cases where *x* is a negative number, [latex]\sqrt{x^{2}}\neq{x}[/latex]! However, in all cases [latex]\sqrt{x^{2}}=\left|x\right|[/latex]. You need to consider this fact when simplifying radicals with an even index that contain variables, because by definition [latex]\sqrt{x^{2}}[/latex] is always nonnegative.

### Taking the Square Root of a Radical Expression

When finding the square root of an expression that contains variables raised to an even power, remember that [latex]\sqrt{x^{2}}=\left|x\right|[/latex].

Examples: [latex]\sqrt{9x^{2}}=3\left|x\right|[/latex], and [latex]\sqrt{16{{x}^{2}}{{y}^{2}}}=4\left|xy\right|[/latex]

We will combine this with the square root of a product rule in our next example to simplify an expression with three variables in the radicand.

### Example

Simplify. [latex] \sqrt{{{a}^{3}}{{b}^{5}}{{c}^{2}}}[/latex]

### Analysis of the Solution

Why did we not write [latex]b^2[/latex] as [latex]|b^2|[/latex]? Because when you square a number, you will always get a positive result, so the principal square root of [latex]\left(b^2\right)^2[/latex] will always be non-negative. One tip for knowing when to apply the absolute value after simplifying any even indexed root is to look at the final exponent on your variable terms. If the exponent is odd – including [latex]1[/latex] – add an absolute value. This applies to simplifying any root with an even index, as we will see in later examples.

In the following video, you will see more examples of how to simplify radical expressions with variables.

We will show another example where the simplified expression contains variables with both odd and even powers.

### Example

Simplify. [latex] \sqrt{9{{x}^{6}}{{y}^{4}}}[/latex]

In our next example, we will start with an expression written with a rational exponent. You will see that you can use a similar process – factoring and sorting terms into squares – to simplify this expression.

### Example

Simplify. [latex] {{(36{{x}^{4}})}^{\frac{1}{2}}}[/latex]

Here is one more example with perfect squares.

### Example

Simplify. [latex] \sqrt{49{{x}^{10}}{{y}^{8}}}[/latex]

## Simplify Cube Roots

We can use the same techniques we have used for simplifying square roots to simplify higher order roots. For example, to simplify a cube root, the goal is to find factors under the radical that are perfect cubes so that you can take their cube root. We no longer need to be concerned about whether we have identified the principal root since we are now finding cube roots. Focus on finding identical trios of factors as you simplify.

### Example

Simplify. [latex] \sqrt[3]{40{{m}^{5}}}[/latex]

Remember that you can take the cube root of a negative expression. In the next example, we will simplify a cube root with a negative radicand.

### Example

Simplify. [latex] \sqrt[3]{-27{{x}^{4}}{{y}^{3}}}[/latex]

You could check your answer by performing the inverse operation. If you are right, when you cube [latex] -3xy\sqrt[3]{x}[/latex] you should get [latex] -27{{x}^{4}}{{y}^{3}}[/latex].

[latex] \begin{array}{l}\left( -3xy\sqrt[3]{x} \right)\left( -3xy\sqrt[3]{x} \right)\left( -3xy\sqrt[3]{x} \right)\\-3\cdot -3\cdot -3\cdot x\cdot x\cdot x\cdot y\cdot y\cdot y\cdot \sqrt[3]{x}\cdot \sqrt[3]{x}\cdot \sqrt[3]{x}\\-27\cdot {{x}^{3}}\cdot {{y}^{3}}\cdot \sqrt[3]{{{x}^{3}}}\\-27{{x}^{3}}{{y}^{3}}\cdot x\\-27{{x}^{4}}{{y}^{3}}\end{array}[/latex]

You can also skip the step of factoring out the negative one once you are comfortable with identifying cubes.

### Example

Simplify. [latex] \sqrt[3]{-24{{a}^{5}}}[/latex]

In the following video, we show more examples of simplifying cube roots.

## Simplifying Fourth Roots

Now let us move to simplifying fourth degree roots. No matter what root you are simplifying, the same idea applies: find cubes for cube roots, powers of four for fourth roots, etc. Recall that when your simplified expression contains an even indexed radical and a variable factor with an odd exponent, you need to apply an absolute value.

### Example

Simplify. [latex] \sqrt[4]{81{{x}^{8}}{{y}^{3}}}[/latex]

An alternative method to factoring is to rewrite the expression with rational exponents, then use the rules of exponents to simplify. You may find that you prefer one method over the other. Either way, it is nice to have options. We will show the last example again, using this idea.

### Example

Simplify. [latex] \sqrt[4]{81{{x}^{8}}{{y}^{3}}}[/latex]

In the following video, we show another example of how to simplify a fourth and fifth root.

For our last example, we will simplify a more complicated expression, [latex]\dfrac{10{{b}^{2}}{{c}^{2}}}{c\sqrt[3]{8{{b}^{4}}}}[/latex]*.** *This expression has two variables, a fraction, and a radical. Let us take it step-by-step and see if using fractional exponents can help us simplify it.

We will start by simplifying the denominator since this is where the radical sign is located. Recall that an exponent in the denominator of a fraction can be rewritten as a negative exponent.

### Example

Simplify. [latex]\dfrac{10{{b}^{2}}{{c}^{2}}}{c\sqrt[3]{8{{b}^{4}}}}[/latex]

Well, that took a while, but you did it. You applied what you know about fractional exponents, negative exponents, and the rules of exponents to simplify the expression.

In our last video, we show how to use rational exponents to simplify radical expressions.

## Summary

A radical expression is a mathematical way of representing the *n*th root of a number. Square roots and cube roots are the most common radicals, but a root can be any number. To simplify radical expressions, look for exponential factors within the radical, and then use the property [latex] \sqrt[n]{{{x}^{n}}}=x[/latex] if *n* is odd and [latex] \sqrt[n]{{{x}^{n}}}=\left| x \right|[/latex] if *n* is even to pull out quantities. All rules of integer operations and exponents apply when simplifying radical expressions.

The steps to consider when simplifying a radical are outlined below.

### Simplifying a radical

When working with exponents and radicals:

- If
*n*is odd, [latex] \sqrt[n]{{{x}^{n}}}=x[/latex]. - If
*n*is even, [latex] \sqrt[n]{{{x}^{n}}}=\left| x \right|[/latex]. (The absolute value accounts for the fact that if*x*is negative and raised to an even power, that number will be positive, as will the*n*th principal root of that number.)