Use the properties of exponents to simplify expressions with rational exponents
Be Prepared 8.7
Before you get started, take this readiness quiz.
Add:
If you missed this problem, review Example 1.28.
Be Prepared 8.8
Simplify:
If you missed this problem, review Example 5.18.
Be Prepared 8.9
Simplify:
If you missed this problem, review Example 5.14.
Simplify Expressions with
Rational exponents are another way of writing expressions with radicals. When we use rational exponents, we can apply the properties of exponents to simplify expressions.
The Power Property for Exponents says that when m and n are whole numbers. Let’s assume we are now not limited to whole numbers.
Suppose we want to find a number p such that We will use the Power Property of Exponents to find the value of p.
So But we know also Then it must be that
This same logic can be used for any positive integer exponent n to show that
Rational Exponent
If is a real number and then
The denominator of the rational exponent is the index of the radical.
There will be times when working with expressions will be easier if you use rational exponents and times when it will be easier if you use radicals. In the first few examples, you’ll practice converting expressions between these two notations.
Example 8.26
Write as a radical expression: ⓐⓑⓒ
We want to write each expression in the form
ⓐ
The denominator of the rational exponent is 2, so the index of the radical is 2. We do not show the index when it is 2.
ⓑ
The denominator of the exponent is 3, so the index is 3.
ⓒ
The denominator of the exponent is 4, so the index is 4.
Try It 8.51
Write as a radical expression: ⓐⓑⓒ
Try It 8.52
Write as a radial expression: ⓐⓑⓒ
In the next example, we will write each radical using a rational exponent. It is important to use parentheses around the entire expression in the radicand since the entire expression is raised to the rational power.
Example 8.27
Write with a rational exponent: ⓐⓑⓒ
We want to write each radical in the form
ⓐ
No index is shown, so it is 2. The denominator of the exponent will be 2.
Put parentheses around the entire expression
ⓑ
The index is 3, so the denominator of the exponent is 3. Include parentheses
ⓒ
The index is 4, so the denominator of the exponent is 4. Put parentheses only around the since 3 is not under the radical sign.
Try It 8.53
Write with a rational exponent: ⓐⓑⓒ
Try It 8.54
Write with a rational exponent: ⓐⓑⓒ
In the next example, you may find it easier to simplify the expressions if you rewrite them as radicals first.
Example 8.28
Simplify: ⓐⓑⓒ
ⓐ
Rewrite as a square root.
Simplify.
ⓑ
Rewrite as a cube root.
Recognize 64 is a perfect cube.
Simplify.
ⓒ
Rewrite as a fourth root.
Recognize 256 is a perfect fourth power.
Simplify.
Try It 8.55
Simplify: ⓐⓑⓒ
Try It 8.56
Simplify: ⓐⓑⓒ
Be careful of the placement of the negative signs in the next example. We will need to use the property in one case.
Example 8.29
Simplify: ⓐⓑⓒ
ⓐ
Rewrite as a fourth root.
Simplify.
ⓑ
The exponent only applies to the 16. Rewrite as a fouth root.
Rewrite 16 as
Simplify.
ⓒ
Rewrite using the property
Rewrite as a fourth root.
Rewrite 16 as
Simplify.
Try It 8.57
Simplify: ⓐⓑⓒ
Try It 8.58
Simplify: ⓐⓑⓒ
Simplify Expressions with
We can look at in two ways. Remember the Power Property tells us to multiply the exponents and so and both equal If we write these expressions in radical form, we get
This leads us to the following definition.
Rational Exponent
For any positive integers m and n,
Which form do we use to simplify an expression? We usually take the root first—that way we keep the numbers in the radicand smaller, before raising it to the power indicated.
Example 8.30
Write with a rational exponent: ⓐⓑⓒ
We want to use to write each radical in the form
ⓐ
ⓑ
ⓒ
Try It 8.59
Write with a rational exponent: ⓐⓑⓒ
Try It 8.60
Write with a rational exponent: ⓐⓑⓒ
Remember that The negative sign in the exponent does not change the sign of the expression.
Example 8.31
Simplify: ⓐⓑⓒ
We will rewrite the expression as a radical first using the defintion, This form lets us take the root first and so we keep the numbers in the radicand smaller than if we used the other form.
ⓐ
The power of the radical is the numerator of the exponent, 2. The index of the radical is the denominator of the exponent, 3.
Simplify.
ⓑ We will rewrite each expression first using and then change to radical form.
Rewrite using
Change to radical form. The power of the radical is the numerator of the exponent, 3. The index is the denominator of the exponent, 2.
Simplify.
ⓒ
Rewrite using
Change to radical form.
Rewrite the radicand as a power.
Simplify.
Try It 8.61
Simplify: ⓐⓑⓒ
Try It 8.62
Simplify: ⓐⓑⓒ
Example 8.32
Simplify: ⓐⓑⓒ
ⓐ
Rewrite in radical form.
Simplify the radical.
Simplify.
ⓑ
Rewrite using
Rewrite in radical form.
Simplify the radical.
Simplify.
ⓒ
Rewrite in radical form.
There is no real number whose square root
Try It 8.63
Simplify: ⓐⓑⓒ
Try It 8.64
Simplify: ⓐⓑⓒ
Use the Properties of Exponents to Simplify Expressions with Rational Exponents
The same properties of exponents that we have already used also apply to rational exponents. We will list the Properties of Exponenets here to have them for reference as we simplify expressions.
Properties of Exponents
If a and b are real numbers and m and n are rational numbers, then
We will apply these properties in the next example.
Example 8.33
Simplify: ⓐⓑⓒ
ⓐ The Product Property tells us that when we multiply the same base, we add the exponents.
The bases are the same, so we add the exponents.
Add the fractions.
Simplify the exponent.
ⓑ The Power Property tells us that when we raise a power to a power, we multiply the exponents.
To raise a power to a power, we multiply the exponents.
Simplify.
ⓒ The Quotient Property tells us that when we divide with the same base, we subtract the exponents.
To divide with the same base, we subtract the exponents.
Simplify.
Try It 8.65
Simplify: ⓐⓑⓒ
Try It 8.66
Simplify: ⓐⓑⓒ
Sometimes we need to use more than one property. In the next example, we will use both the Product to a Power Property and then the Power Property.
Example 8.34
Simplify: ⓐⓑ
ⓐ
First we use the Product to a Power Property.
Rewrite 27 as a power of 3.
To raise a power to a power, we multiply the exponents.
Simplify.
ⓑ
First we use the Product to a Power Property.
To raise a power to a power, we multiply the exponents.
Try It 8.67
Simplify: ⓐⓑ
Try It 8.68
Simplify: ⓐⓑ
We will use both the Product Property and the Quotient Property in the next example.
Example 8.35
Simplify: ⓐⓑ
ⓐ
Use the Product Property in the numerator, add the exponents.
Use the Quotient Property, subtract the exponents.
Simplify.
ⓑ Follow the order of operations to simplify inside the parenthese first.
Use the Quotient Property, subtract the exponents.
Simplify.
Use the Product to a Power Property, multiply the exponents.
Try It 8.69
Simplify: ⓐⓑ
Try It 8.70
Simplify: ⓐⓑ
Media Access Additional Online Resources
Access these online resources for additional instruction and practice with simplifying rational exponents.