Simplifying Expressions with Exponents

 

Learning Objective(s)

·         Simplify algebraic expressions with exponents.

 

Introduction

 

Exponents can be attached to variables as well as numbers. When they are, the basic rules of exponents and exponential notation apply when writing and simplifying algebraic expressions that contain exponents.

 

Simple Rules of Exponents

 

Let’s look at some of the basic rules of exponents.

 

Any number or variable raised to a power of one is simply that number.  In return, any number or variable that doesn’t have an exponent shown with it can be considered to have an exponent of 1.  Below are some examples:

 

51 = 5

18 = 181

x1 = x

xy = x1y1

 

Another rule of exponents is that any non-zero number or variable raised to a power of 0 equals 1.   = 1 for x ≠ 0.

 

 

Simplify the expression

                    

A) 0

 

B)

 

C)

 

D)

 

Show/Hide Answer

A) Incorrect. , not 0. The correct answer is C) .

 

B) Incorrect. , not x. The correct answer is C) .

 

C) Correct.  and

 

D) Incorrect.  can be written more simply as y. The correct answer is C) .

 

 

As with numbers, variables raised to a negative power are equivalent to 1 over the variable to the same but positive power.  For example:

 

 

 

 

Exponent of 0 or 1 and Negative Exponents

 

Any number or variable raised to a power of 1 is the number itself.  

 

Any non-zero number or variable raised to a power of 0 is equal to 1.    

 

Any non-zero number n and any integer x, . For example,

 

Notice that these rules say that the base, n, must be a “nonzero number”. When n is 0, both n0 and n raised to a negative power are undefined.

 

 

Let’s take a look at how to simplify an algebraic expression with negative exponents in the numerator and the denominator. It’s helpful, especially at first to separate the variables and simplify them individually:

 

Example

 

Problem

Simplify

 

 

 

 

 

Rewrite as a product of fractions

 

 

Rewrite variables with negative powers following the rule for negative exponents:

 

 

 

Simplify division by a fraction

 

 

 

Multiply fractions

 

Answer

 

 

 

 

Products and Quotients of Powers

 

Now that we’ve seen how to simplify variables with exponents, let’s learn how to multiply and divide them.

 

We’ll start with finding the product of two exponential terms with the same base. To find the product of powers with the same base, just add the exponents and keep the base the same. Consider the example.  We could rewrite this problem as follows:

 

 =  =  =

 

Or we could just add the exponents:

 

 =  =

 

That second method is a lot easier.

 

Division with exponential notation has a similar short cut. To find the quotient of powers that have the same base, subtract the exponents. Here’s how it works:

 

Example

 

Problem

Simplify

 

 

 

 

Rewrite exponential notation

 

Rewrite as product of fractions

 

Use associative property to regroup factors

 

Because 

 

Product of powers

Answer

 

 

 

 

Here are a few additional examples:

 

 

 

Rules for Multiplying and Dividing with Exponents

 

When multiplying two terms with the same base, add the exponents:

 

When dividing two terms with the same base, subtract the exponent in the denominator from the exponent in the numerator:

 

Note: 00 is undefined, so these rules do not apply in that situation.

 

 

 

Simplify the expression

 

A)

 

B)

 

C)

 

D)

 

Show/Hide Answer

A) Incorrect. You subtracted the numbers, but that rule only applies to the exponents. You must divide 30 by 5. The correct answer is D) .

 

B) Incorrect. When dividing, you subtract, not add, exponents. The correct answer is D) .

 

C) Incorrect. When dividing, you subtract, not divide, exponents. The correct answer is D) .

 

D) Correct. When dividing terms in exponential form, you divide the coefficients and subtract the exponents of terms with the same base.

 

 

 

Power of a Power

 

Exponential notation is a shorter way of writing repeated multiplication.  When a variable expression with exponents is raised to a power, we could apply the rules for multiplying powers:

 

 

Example

Problem

Simplify

 

 

 

Write as factors

Regroup

 

Add exponents with the same base

Answer

 

 

 

 

That worked, but it was cumbersome. There’s an easier way. Notice that the final exponent of each variable was the product of the exponent inside the parentheses, 1, and the exponent outside the parentheses, 3. To find the power of a power, multiply the exponents.

 

Let’s try it again:

 

Example

Problem

Simplify  

 

 

Multiply the exponents

 

 

Simplify

 

 

Simplify

 

Answer

 

 

 

That was quicker!

 

Power of a Power:

 

To raise a power to a power, multiply the exponents.

 

 

 

Simplify

 

A)

 

B)

 

C)

 

D)

 

Show/Hide Answer

A) Incorrect. 33 = 27. The correct answer is B) .

 

B) Correct. When you are finding a power of a power, you multiply exponents.

 

C) Incorrect.   When you are finding the power of a power, you multiply, not add, exponents. The correct answer is B) .

 

D) Incorrect. 33 = 27, and when you are finding a power of a power, you multiply exponents, not write them next to one another. The correct answer is B) .

 

 

Summary

 

The rules of exponents apply to both numbers and variables. In brief, these rules are as follows:

 

Exponent of 0 or 1 and Negative Exponents

 

Any number or variable raised to a power of 1 is the number itself.  

 

Any non-zero number or variable raised to a power of 0 is equal to 1. 

 

Any non-zero number n and any integer x, . For example,

 

Product of a Power

 

When multiplying two terms with the same base, add the exponents:

 

Quotient of a Power

When dividing two terms with the same base, subtract the exponent in the denominator from the exponent in the numerator:

 

Power of a Power:

 

To raise a power to a power, multiply the exponents.

 

The rules of exponents provide accurate and efficient shortcuts for simplifying variables in exponential notation.