Rules of Exponents

 

Learning Objective(s)

·         Understand rules of exponents.

·         Simplify and solve expressions in exponential notation.

 

Introduction

 

We need a common language in order to communicate mathematical ideas clearly and efficiently. Exponential notation is one example. It was developed to express repeated multiplication and to make it easier to write very long numbers. For example, growth models for populations often use exponents to manage and manipulate large numbers that change quickly over time.  

 

In order to work with exponents, we need to “speak the language” and learn a few rules first.

 

What is Exponential Notation?

 

Exponential notation has two parts. The base, as the name suggests, is the number on the bottom. The other part of the notation is a small number written in superscript to the right of the base, called the exponent.  Below are some examples of exponential notation. We’ll use these examples to learn about the notation.

 

103

251

-34

 

Let’s start with 103. The base is 10. This means that 10 is a factor, and it’s going to be multiplied by itself some number of times. The precise number of times is given by the exponent, the number in superscript. In this case, the exponent is 3, which means the base of 10 will be used a factor 3 times. So 103 means 10 • 10 • 10.

 

Now we know what 103 means, but how do we pronounce it? We have a lot of choices: this term could be said as “10 raised to the third power” or “10 to the third,” or “10 cubed.”  The words “raised to a power” are inserted between the base and the exponent to indicate exponential notation.

 

Okay then. Let’s consider 251. What does an exponent of 1 mean? Any value raised to a power of 1 is just the value. This makes sense when we think about it, because the exponent of 1 means the base is used as a factor only once. So the base stands alone, and 251 is simply 25.

 

That leaves us with the term -34. This example is a little trickier because there is a negative sign in there. One of the rules of exponential notation is that the exponent relates only to the value immediately to its left. So, -34 does not mean -3 • -3 • -3 • -3. It means “the opposite of 34,” or — (3 • 3 • 3 • 3). If we wanted the base to be -3, we’d have to use parentheses in the notation: (-3)4.  Why so picky? Well, do the math:

 

-34 = – (3 • 3 • 3 • 3) = -81

(-3)4 = -3 • -3 • -3 • -3 = 81

 

That’s an important difference.

 

Rules for Computing with Exponents

We just learned the rule that the exponent only relates to the number directly to the left unless parentheses are used. Here’s another rule—when an exponent is present outside parentheses, everything inside is raised to that power. Consider the following example:

(5 + 3)2

According to the order of operations, we must first simplify what is in the parentheses before we do any other operations. So we add 5 and 3 and then square the sum, 8, to arrive at an answer of 64. Another way to proceed is to rewrite (5 + 3)2 as (5 + 3)(5 + 3), and then multiply it out to again get 64.

(5 + 3)2 = (8)2 = 8 • 8 = 64

 

(5 + 3)2 = (5 + 3)(5 + 3) = 5(5 + 3) + 3(5 +3) = 25 + 15 + 15 + 9 = 64

 

Parenthesis can be used in others ways with exponential notation. For example, we can use them to describe an exponential term to a power. For example, let’s take 52 and raise it to the 4th power. We’d write that as (52)4. When a number written in exponential notation is raised to a power, it is called a “power of a power.”

 

In this expression, the base is 52 and the exponent is 4: 52 is to be used as a factor 4 times. We could rewrite this problem 52 • 52 • 52 • 52 or (5 • 5) • (5 • 5) • (5 • 5) • (5 • 5).  Notice that works out to 5 multiplied 8 times. What’s another way to write that? 58.

 

That leads us to another rule. Compare 58 to the original term of (52)4. Notice that the new exponent is the same as the product of the original exponents: 2 • 4 = 8. A short cut for simplifying the power of a power is to multiply the exponents and keep the base the same.

 

There’s also a rule for combining two numbers in exponential form that have the same base. Consider the following expression:

(23)(24)

This can be rewritten as (2 • 2 • 2) (2 • 2 • 2 • 2) or  2 • 2 • 2 • 2 • 2 • 2 • 2. In exponential form, you would write the product as 27. Notice 7 is the sum of the original two exponents, 3 and 4. To multiply exponential terms with the same base, add the exponents.

 

Rules of Exponents

 

An exponent applies only to the value to its immediate left.

 

When a quantity in parentheses is raised to a power, the exponent applies to everything inside the parentheses.

 

To multiply two terms with the same base, add their exponents. (nx)(ny)=nx+y

 

To raise a power to a power, multiply the exponents. (nx)y= nxy

 

 

 

Simplify the expression, keeping the answer in exponential notation.

 

(23 • 22)4

 

A) 224

B) 49

C) 220

D) 29

 

Show/Hide Answer

A. Incorrect. 224 is not equivalent to (23 • 22)4. Only multiply exponents when taking the power of a power, not when you are multiplying terms. Then, you add the exponents. So the quantity in the parentheses is 25, not 26. The correct answer is 220.

 

B. Incorrect. 49 is not equivalent to (23 • 22)4. When you multiply powers, you add the exponents but keep the base the same. The quantity in the parentheses is 25. Then to raise this to a power of 4, you multiply the exponents 5 and 4. The correct answer is 220.

 

C. Correct.  (23 • 22)4 = 220. First, simplify inside the parentheses: 23 • 22 = 25 (add the exponents). Then raise that term to the power outside the parentheses: (25)4 = 220 (multiply the exponents).

 

D. Incorrect. 29 is not equivalent to (23 • 22)4. In this incorrect response, all the exponents were added, but when you took the power of a power, you should have multiplied exponents. (23 • 22)4= (25)4 = 220 The correct answer is 220.

 

 

Zero and Negative Exponents

 

Exponents are not always positive. But what does it mean when an exponent is 0 or a negative integer? Let’s use what we know about powers of 10 to find out. Below is a list of powers of 10 and their equivalent values. Look at how the numbers change going down the left and right columns. There’s a pattern there—see it?

 

Exponential Form

Expanded Form

Value

105

10  • 10 • 10 • 10 • 10

100,000

104

10  • 10 • 10 • 10

10,000

103

10  • 10 • 10

1,000

102

10 • 10

100

101

10

10

 

Moving down the table, each row drops one factor of 10 from the one above it. From row 1 to row 2, the exponential form goes from 105 to 104. The value drops from 100,000 to 10,000. Another way to put this is that each value is divided by 10 to produce the next value down the column.

 

We can continue this pattern to add some more rows as shown below, each time dividing the number in one row by 10 to get the number in the next row: 

 

Exponential Form

Expanded Form

Value

105

10  • 10 • 10 • 10 • 10

100,000

104

10  • 10 • 10 • 10

10,000

103

10  • 10 • 10

1,000

102

10 • 10

100

101

10

10

100

1

1

10-1

0.1

 

10-2

 

0.01

 

Following the pattern, we see that 100 is equal to 1. Then we get into negative exponents: 10-1 is equal to , and 10-2 is the same as . Now that’s interesting. Look back at the table, and see what 10 is in exponential form. It’s 101. If we substitute that form of 10 into the fraction , the fraction becomes . So 10-1 =. Something very similar can be done with 10-2:

 

10-2 =  and 100 = 102

 

 

10-2 =

 

How about that? Numbers with negative exponents can be rewritten as a fraction, and not just any fraction. A number raised to a negative power is equivalent to the reciprocal of the number raised to the opposite of the power. That sounds complicated, but all it means is what we’ve just seen. A number to a negative power is the same as 1 over the number to the same but positive power.  For example, 10-3 = and 10-7=.

 

To see if these patterns hold true for numbers other than 10, check out a table with powers of 3:

 

Exponential Form

Expanded Form

Value

35

3 • 3 • 3 • 3 • 3

243

34

3 • 3 • 3 • 3

81

33

3 • 3 • 3

27

32

3 • 3

9

31

3

3

30

1

1

3-1

3-2

 

 

Yes, it all looks the same. The numbers are different but the patterns are the same. Now we know how numbers with zero and negative exponents behave.

 

More Rules of Exponents

 

For any nonzero number n, n0 = 1. For example, 180 = 1.

For any nonzero number n and any integer x, n-x= . For example, 5-2= .

 

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.

 

 

 

Luisa and Michele are working together to simplify the following exponential expression:

 

(3 + 2)2 • (53)2 • (50)( 54)

 

They discover that they have different solutions:

 

Luisa’s answer: 32 • 22 • 56 • 54 = 32 • 22 • 510

Michele answer: 52 • 55 • 0 = 0

 

Which of the girls has used the rules of exponents properly to get a correct and simplified exponential expression?

 

A) Both Luisa’s and Michele’s answers are correct and equivalent to the original expression.

B) Neither Luisa’s nor Michele’s answer is correct and equivalent to the original expression.

C) Only Luisa has correctly simplified the original expression.

D) Only Michelle has correctly simplified the original expression.

 

Show/Hide Answer

A) Incorrect. Luisa has incorrectly simplified the first term in the expression. (3 + 2)2 = 52 or 25, not 32 • 22, which is 9 • 4 or 36. Michele has made two errors in her work. First, to simplify a power of a power, she should have multiplied, not added, exponents: (53)2 = 56. Second, (50)(54) = 54, because 50 = 1 or because (50)(54) = 50 + 4. The correct answer is B, they’re both wrong.

 

B) Correct. Neither girl has simplified correctly. Luisa has incorrectly simplified the first term in the expression. (3 + 2)2 = 52 or 25, 32 • 22, which is 9 • 4 or 36. Michele has made two errors in her work. First, to simplify a power of a power, she should have multiplied, not added, exponents: (53)2 = 56. Second, (50)(54) = 54, because 50 = 1 or because (50)(54) = 50 + 4.  (This expression can be simplified to 512.)

 

C) Incorrect. Luisa has incorrectly simplified the first term in the expression. (3 + 2)2 = 52 or 25, not 32 • 22, which is 9 • 4 or 36. The correct answer is B, they are both incorrect.

 

D) Incorrect. Michele has made two errors in her work. First, to simplify a power of a power, she should have multiplied, not added, exponents: (53)2 = 56. Second, (50)(54)= 54, because 50 = 1 or because (50)(54) = 50 + 4. The correct answer is B, neither of them is correct.

 

 

Summary

 

Exponential notation is composed of a base and an exponent. It is a “shorthand” way of writing repeated multiplication, and indicates that the base is a factor and the exponent is the number of times the factor is used in the multiplication. The basic rules of exponents are as follows:

 

An exponent applies only to the value to its immediate left.

 

When a quantity in parentheses is raised to a power, the exponent applies to everything inside the parentheses.

 

To multiply two terms with the same base, add their exponents. (nx)(ny)=nx+y

 

To raise a power to a power, multiply the exponents. (nx)y= nxy

 

For any nonzero number n, n0 = 1.

For any nonzero number n and any integer x, n-x=.