# Roots and Radicals

## Rational Exponents


### 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.



But we know also . Then it must be that .
This same logic can be used for any positive integer exponent n to show that .
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.
In the next example, you may find it easier to simplify the expressions if you rewrite them as radicals first.
Be careful of the placement of the negative signs in the next example. We will need to use the property  in one case.

### Simplify Expressions with 

Let’s work with the Power Property for Exponents some more.
Suppose we raise  to the power m.


Now suppose we take  to the  power.



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.
Remember that . The negative sign in the exponent does not change the sign of the expression.

### Use the Laws of Exponents to Simplify Expressions with Rational Exponents

The same laws of exponents that we already used apply to rational exponents, too. We will list the Exponent Properties here to have them for reference as we simplify expressions.
When we multiply the same base, we add the exponents.
We will use the Power Property in the next example.
The Quotient Property tells us that when we divide with the same base, we subtract the exponents.
Sometimes we need to use more than one property. In the next two examples, we will use both the Product to a Power Property and then the Power Property.
We will use both the Product and Quotient Properties in the next example.

### Key Concepts

1. Summary of Exponent Properties
2. If  are real numbers and  are rational numbers, then



### Section Exercises


### Practice Makes Perfect

Simplify Expressions with 
In the following exercises, write as a radical expression.
In the following exercises, write with a rational exponent.
In the following exercises, simplify.
Simplify Expressions with 
In the following exercises, write with a rational exponent.
In the following exercises, simplify.
Use the Laws of Exponents to Simplify Expressions with Rational Exponents
In the following exercises, simplify.

### Everyday Math


### Writing Exercises


### Chapter 9 Review Exercises


### Simplify and Use Square Roots

Simplify Expressions with Square Roots
In the following exercises, simplify.
Estimate Square Roots
In the following exercises, estimate each square root between two consecutive whole numbers.
Approximate Square Roots
In the following exercises, approximate each square root and round to two decimal places.
Simplify Variable Expressions with Square Roots
In the following exercises, simplify.

### Simplify Square Roots

Use the Product Property to Simplify Square Roots
In the following exercises, simplify.
Use the Quotient Property to Simplify Square Roots
In the following exercises, simplify.

### Add and Subtract Square Roots

Add and Subtract Like Square Roots
In the following exercises, simplify.
Add and Subtract Square Roots that Need Simplification
In the following exercises, simplify.

### Multiply Square Roots

Multiply Square Roots
In the following exercises, simplify.
Use Polynomial Multiplication to Multiply Square Roots
In the following exercises, simplify.

### Divide Square Roots

Divide Square Roots
In the following exercises, simplify.
Rationalize a One Term Denominator
In the following exercises, rationalize the denominator.
Rationalize a Two Term Denominator
In the following exercises, rationalize the denominator.

### Solve Equations with Square Roots

Solve Radical Equations
In the following exercises, solve the equation.
Use Square Roots in Applications
In the following exercises, solve. Round approximations to one decimal place.

### Higher Roots

Simplify Expressions with Higher Roots
In the following exercises, simplify.
Use the Product Property to Simplify Expressions with Higher Roots
In the following exercises, simplify.
Use the Quotient Property to Simplify Expressions with Higher Roots
In the following exercises, simplify.
Add and Subtract Higher Roots
In the following exercises, simplify.

### Rational Exponents

Simplify Expressions with 
In the following exercises, write as a radical expression.
In the following exercises, write with a rational exponent.
In the following exercises, simplify.
Simplify Expressions with 
In the following exercises, write with a rational exponent.
In the following exercises, simplify.
Use the Laws of Exponents to Simplify Expressions with Rational Exponents
In the following exercises, simplify.

### Practice Test

In the following exercises, simplify.
In the following exercises, rationalize the denominator.
In the following exercises, solve.
In the following exercise, solve.