## 9.2 – Radical Expressions and Rational Exponents

### Learning Objectives

• (9.2.1) – Define and identify a radical expression
• (9.2.2) – Convert radicals to expressions with rational exponents
• (9.2.3) – Convert expressions with rational exponents to their radical equivalent
• (9.2.4) – Rational exponents whose numerator is not equal to one
• (9.2.5) – Simplify Radical Expressions
• Simplify radical expressions using factoring
• Simplify radical expressions using rational exponents and the laws of exponents

# (9.2.1) – Define and identify a radical expression

Square roots are most often written using a radical sign, like this, $\sqrt{4}$. But there is another way to represent them. You can use rational exponents instead of a radical. A rational exponent is an exponent that is a fraction. For example, $\sqrt{4}$ can be written as ${{4}^{\tfrac{1}{2}}}$.

Can’t imagine raising a number to a rational exponent? They may be hard to get used to, but rational exponents can actually help simplify some problems. Writing radicals with rational exponents will come in handy when we discuss techniques for simplifying more complex radical expressions.

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

# (9.2.2) – Convert radicals to expressions with rational exponents

Radicals and fractional exponents are alternate ways of expressing the same thing.  In the table below we show equivalent ways to express radicals: with a root, with a rational exponent, and as a principal root.

Exponent Form

Principal Root

$\sqrt{16}$ ${{16}^{\tfrac{1}{2}}}$ 4
$\sqrt{25}$ ${{25}^{\tfrac{1}{2}}}$ 5
$\sqrt{100}$ ${{100}^{\tfrac{1}{2}}}$ 10

Let’s look at some more examples, but this time with cube roots. Remember, cubing a number raises it to the power of three. Notice that in the examples in the table below, the denominator of the rational exponent is the number 3.

Exponent Form

Principal Root

$\sqrt{8}$ ${{8}^{\tfrac{1}{3}}}$ 2
$\sqrt{8}$ ${{125}^{\tfrac{1}{3}}}$ 5
$\sqrt{1000}$ ${{1000}^{\tfrac{1}{3}}}$ 10

These examples help us model a relationship between radicals and rational exponents: namely, that the $n^{th}$ root of a number can be written as either $\sqrt[n]{x}$ or ${{x}^{\frac{1}{n}}}$.

Exponent Form

$\sqrt{x}$ ${{x}^{\tfrac{1}{2}}}$
$\sqrt{x}$ ${{x}^{\tfrac{1}{3}}}$
$\sqrt{x}$ ${{x}^{\tfrac{1}{4}}}$
$\sqrt[n]{x}$ ${{x}^{\tfrac{1}{n}}}$

In the table above, notice how the denominator of the rational exponent determines the index of the root. So, an exponent of $\frac{1}{2}$ translates to the square root, an exponent of $\frac{1}{5}$ translates to the fifth root or ${\,}^5\hspace{-0.1in} \sqrt{\,\,\,}$, and $\frac{1}{8}$ translates to the eighth root or ${\,}^8\hspace{-0.1in} \sqrt{\,\,\,}$.

### Example

Express ${{(2x)}^{^{\frac{1}{3}}}}$ in radical form.

Remember that exponents only refer to the quantity immediately to their left unless a grouping symbol is used. The example below looks very similar to the previous example with one important difference—there are no parentheses! Look what happens.

### Example

Express $2{{x}^{^{\frac{1}{3}}}}$ in radical form.

# (9.2.3) – Convert expressions with rational exponents to their radical equivalent Flexibility

We can write radicals with rational exponents, and as we will see when we simplify more complex radical expressions, this can make things easier. Having different ways to express and write algebraic expressions allows us to have flexibility in solving and simplifying them. It is like having a thesaurus when you write, you want to have options for expressing yourself!

### Example

Write $\sqrt{81}$ as an expression with a rational exponent.

### Example

Express $4\sqrt{xy}$ with rational exponents.

# (9.2.4) – Rational exponents whose numerator is not equal to one

All of the numerators for the fractional exponents in the examples above were 1. You can use fractional exponents that have numerators other than 1 to express roots, as shown below.

Exponent

$\sqrt{9}$ $9^{\frac{1}{2}}$
$\sqrt{{{9}^{2}}}$ $9^{\frac{2}{3}}$
$\sqrt{9^{3}}$ $9^{\frac{3}{4}}$
$\sqrt{9^{2}}$ $9^{\frac{2}{5}}$
$\sqrt[n]{9^{x}}$ $9\frac{x}{n}$ To rewrite a radical using a fractional exponent, the power to which the radicand is raised becomes the numerator and the root/ index becomes the denominator.

### Writing Rational Exponents

Any radical in the form $\large \sqrt[n]{a^{m}}$  can be written using a fractional exponent in the form $\large a^{\frac{m}{n}}$.

The relationship between $\sqrt[n]{{{a}^{m}}}$and ${{a}^{\frac{m}{n}}}$ works for rational exponents that have a numerator of 1 as well. For example, the radical $\sqrt{8}$ can also be written as $\sqrt{{{8}^{1}}}$, since any number remains the same value if it is raised to the first power. You can now see where the numerator of 1 comes from in the equivalent form of ${{8}^{\frac{1}{3}}}$.

In the next example, we practice writing radicals with rational exponents where the numerator is not equal to one.

### Example

1. $\sqrt{{{a}^{6}}}$
2. $\sqrt{16^3}$

In our last example we will rewrite expressions with rational exponents as radicals. This practice will help us when we simplify more complicated radical expressions, and as we learn how to solve radical equations. Typically it is easier to simplify when we use rational exponents, but this exercise is intended to help you understand how the numerator and denominator of the exponent are the exponent of a radicand and index of a radical.

### Example

Rewrite the expressions using a radical.

1. ${x}^{\frac{2}{3}}$
2. ${5}^{\frac{4}{7}}$

In the following video we show more examples of writing radical expressions with rational exponents and expressions with rational exponents as radical expressions.

We will use this notation later, so come back for practice if you forget how to write a radical with a rational exponent.

# (9.2.5) – Simplify Radical Expressions

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

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 $\left(ab\right)^{x}=a^{x}\cdot{b}^{x}$. So, for example, you can use the rule to rewrite ${{\left( 3x \right)}^{2}}$ as ${{3}^{2}}\cdot {{x}^{2}}=9\cdot {{x}^{2}}=9{{x}^{2}}$.

Now instead of using the exponent 2, let’s use the exponent $\frac{1}{2}$. The exponent is distributed in the same way.

${{\left( 3x \right)}^{\frac{1}{2}}}={{3}^{\frac{1}{2}}}\cdot {{x}^{\frac{1}{2}}}$

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

$\sqrt{3x}=\sqrt{3}\cdot \sqrt{x}$

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 $b$, $\sqrt{ab}=\sqrt{a}\cdot \sqrt{b}$.

For example: $\sqrt{100}=\sqrt{10}\cdot \sqrt{10}$, and $\sqrt{75}=\sqrt{25}\cdot \sqrt{3}$

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 $\sqrt{(2\cdot 2)(2\cdot 2)(3\cdot 3})$, you can rewrite the expression as the product of multiple perfect squares: $\sqrt{{{2}^{2}}}\cdot \sqrt{{{2}^{2}}}\cdot \sqrt{{{3}^{2}}}$.

The square root of a product rule will help us simplify roots that aren’t perfect, as is shown the following example.

### Example

Simplify. $\sqrt{63}$

The final answer $3\sqrt{7}$ 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 $\sqrt{{{x}^{2}}}$. This looks like it should be equal to x, right? Let’s 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 $\sqrt{{{x}^{2}}}$. Where are they equal? Where are they not equal?

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

$x$ $x^{2}$ $\sqrt{x^{2}}$ $\left|x\right|$
$−5$ 25 5 5
$−2$ 4 2 2
0 0 0 0
6 36 6 6
10 100 10 10

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

### Taking the Square Root of a Radical Expression

When finding the square root of an expression that contains variables raised to a power, consider that $\sqrt{x^{2}}=\left|x\right|$.

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

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. $\sqrt{{{a}^{3}}{{b}^{5}}{{c}^{2}}}$

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. $\sqrt{9{{x}^{6}}{{y}^{4}}}$

### ExAMPLE

Simplify $\sqrt{x^2-6x+9}$.

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. ${{(36{{x}^{4}})}^{\frac{1}{2}}}$

Here is one more example with perfect squares.

### Example

Simplify. $\sqrt{49{{x}^{10}}{{y}^{8}}}$

#### 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. $\sqrt{40{{m}^{5}}}$

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. $\sqrt{-27{{x}^{4}}{{y}^{3}}}$

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

### Example

Simplify. $\sqrt{-24{{a}^{5}}}$

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

#### Simplifying fourth roots

Now let’s 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. $\sqrt{81{{x}^{8}}{{y}^{3}}}$

### Simplify radical expressions using rational exponents and the laws of exponents

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. $\sqrt{81{{x}^{8}}{{y}^{3}}}$

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, $\large\frac{10{{b}^{2}}{{c}^{2}}}{c\sqrt{8{{b}^{4}}}}$. This expression has two variables, a fraction, and a radical. Let’s 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 or a fraction can be rewritten as a negative exponent.

### Example

Simplify. $\displaystyle \frac{10{{b}^{2}}{{c}^{2}}}{c\sqrt{8{{b}^{4}}}}$

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 nth 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 $\sqrt[n]{{{x}^{n}}}=x$ if n is odd, and $\sqrt[n]{{{x}^{n}}}=\left| x \right|$ 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.

• If n is odd, $\sqrt[n]{{{x}^{n}}}=x$.
• If n is even, $\sqrt[n]{{{x}^{n}}}=\left| x \right|$. (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 nth principal root of that number.)