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Beginning Algebra
Tutorial 26: Exponents

Learning Objectives

 After completing this tutorial, you should be able to: Use the definition of exponents. Simplify exponential expressions involving multiplying like bases, zero as an exponent, dividing like bases, raising a base to two exponents, raising a product to an exponent and raising a quotient to an exponent.

Introduction

 This tutorial covers the basic definition and some of the rules of exponents.  The rules it covers are the product rule, quotient rule, power rule, power of a product rule and power of a quotient rule as well as the definitions for zero and negative exponents. Exponents are everywhere in algebra and beyond.  Let's see what we can do with exponents.

Tutorial

 Definition of Exponents (note there are n  x's in the product) x = base,    n = exponent

 Exponents are another way to write multiplication. The exponent tells you how many times a base appears in a PRODUCT.     Example 1: Evaluate .

 *Write the base 1/4 in a product 3 times *Multiply

 Example 2:   Evaluate .

 *Write the base -6 in a product 2 times *Multiply

 Note how I included the - when I expanded this problem out.  If the - is inside the ( ) of an exponent, then it is included as part of the base.

 Example 3:   Evaluate .

 *Negate 6 squared *Put a - in front of 6 written in a product 2 times *Multiply

 Hey, this looks a lot like example 2!!!!  It may look alike, but they ARE NOT exactly the same.  Can you see the difference between the two??  Hopefully, you noticed that in example 2, there was a ( ) around the - and the 6.  In this problem, there is no -.  This means the - is NOT part of the base, so it will not get expanded like it did in example 2.  It is interpreted as finding the negative or opposite of 6 squared.

 Multiplying Like Bases With Exponents (The Product Rule for Exponents) Specific Illustration

 Let’s first start by using the definition of exponents to help you to understand how we get to the law for multiplying like bases with exponents: Note that 2 + 3 = 5, which is the exponent we ended up with.  We had 2 x’s written in a product plus another 3 x’s written in the product for a total of 5 x’s in the product.  To indicate that we put the 5 in the exponent.   Let's put this idea together into a general rule:

 Multiplying Like Bases With Exponents (The Product Rule for Exponents) in general,

 In other words, when you multiply like bases you add your exponents.  The reason is, exponents count how many of your base you have in a product, so if you are continuing that product, you are adding on to the exponents.   Example 4:   Use the product rule to simplify the expression .

 *When mult. like bases you add your exponents

 Example 5:   Use the product rule to simplify the expression .

 *When mult. like bases you add your exponents

 Zero as an exponent

 Except for 0, any base raised to the 0 power simplifies to be the number 1. Note that the exponent doesn’t become 1, but the whole expression simplifies to be the number 1.     Example 6:  Evaluate .

 *Any expression raised to the 0 power simplifies to be 1

 Example 7:  Evaluate .

 Be careful on this example.  The order of operations shown in Tutorial 4: Introduction to Variable Expressions and Equations says to evaluate exponents before doing any multiplication.  This means we need to find x raised to the 0 power first and then multiply it by 3.

 *x raised to the 0 power is 1 *Multiply

 Dividing Like Bases With Exponents (Quotient Rule for Exponents) Specific Illustration

 Let’s first start by using the definition of exponents to help you to understand how we get to the law for dividing like bases with exponents: Note how 5 - 2 = 3, the final answer’s exponent.  When you multiply you are adding on to your exponent, so it should stand to reason that when you divide like bases you are taking away from your exponent. Let's put this idea together into a general rule:

 Dividing Like Bases With Exponents (Quotient Rule for Exponents) in general,

 In other words, when you divide like bases you subtract their exponents. Keep in mind that you always take the numerator’s exponent minus your denominator’s exponent, NOT the other way around.   Example 8:  Use the quotient rule to simplify the expression .

 *When div. like bases you subtract your exponents

 Example 9:  Use the quotient rule to simplify the expression .

 *When div. like bases you subtract your exponents

 Base Raised to Two Exponents (Power Rule for Exponents) Specific Illustration

 Let’s first start by using the definition of exponents as well as the law for multiplying like bases  to help you to understand how we get to the law for raising a base to two exponents: Note how 2 times 3 is 6, which is the exponent of the final answer.   We can think of this as 3 groups of 2, which of course would come out to be 6.

 Base Raised to two Exponents (Power Rule for Exponents) in general,

 In other words, when you raise a base to two exponents, you multiply those exponents together. Again, you can think of it as n groups of m if it helps you to remember.   Example 10:   Use the power rule for exponents to simplify the expression .

 *When raising a base to 2 powers you mult. your exponents

 A Product Raised to an Exponent (Power of a Product Rule) Specific Illustration

 Let’s first start by using the definition of exponents to help you to understand how we get to the law for raising a product to an exponent: Note how both bases of your product ended up being raised by the exponent of 3.

 A Product Raised to an Exponent (Power of a Product Rule) in general,

 In other words, when you have a PRODUCT (not a sum or difference) raised to an exponent, you can simplify by raising each base in the product to that exponent.   Example 11:   Use the power of a product rule to simplify the expression .

 *When raising a product to an exponent, raise each base of the product to that exponent

 A Quotient Raised to an Exponent (Power of a Quotient Rule) Specific Illustration

 Let’s first start by using the definition of exponents to help you to understand how we get to the law for raising a quotient to an exponent: Since, division is really multiplication of the reciprocal, it has the same basic idea as when we raised a product to an exponent.

 A Quotient Raised to an Exponent (Power of a Quotient Rule) in general,

 In other words, when you have a QUOTIENT (not a sum or difference) raised to an exponent, you can simplify by raising each base in the numerator and denominator of the quotient to that exponent.   Example 12:   Use the power of a quotient rule to simplify the expression .

 *When raising a quotient to an exponent, raise each base of the quotient to that exponent *Use def. of exponents to evaluate

 Simplifying an Exponential Expression

 When simplifying an exponential expression,  write it so that each base is written one time with one exponent.  In other words, write it in the most condense form you can. A lot of times you are having to use more than one rule to get the job done.  As long as you are using the rule appropriately, you should be fine.

 Example 13:  Simplify .

 *When mult. like bases you add your exponents     *When div. like bases you subtract your exponents

 Example 14:  Simplify .

 *When raising a product to an exponent, raise each base of the product to that exponent     *When div. like bases you subtract your exponents

Practice Problems

 These are practice problems to help bring you to the next level.  It will allow you to check and see if you have an understanding of these types of problems. Math works just like anything else, if you want to get good at it, then you need to practice it.  Even the best athletes and musicians had help along the way and lots of practice, practice, practice, to get good at their sport or instrument.  In fact there is no such thing as too much practice. To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that  problem.  At the link you will find the answer as well as any steps that went into finding that answer.

Practice Problems 1a - 1e: Simplify.

Need Extra Help on these Topics?

The following are webpages that can assist you in the topics that were covered on this page:

 http://www.purplemath.com/modules/exponent.htm This webpage helps you with the rules of exponents. http://www.sosmath.com/algebra/logs/log2/log2.html#shortcuts This webpage helps you with the definition of exponents. http://www.sosmath.com/algebra/logs/log3/log31/log31.html This webpage helps with the product rule for exponents. http://www.sosmath.com/algebra/logs/log3/log33/log33.html This webpage helps with the rule for raising a base to two exponents.  http://www.ltcconline.net/greenl/courses/152A/polyExp/intexp.htm This webpage goes over the rules of exponents.

Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.

Last revised on August 2, 2011 by Kim Seward.
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