Easy Math STEP-BY-STEP: Master High-Frequency Concepts and Skills for Mathematical Proficiency

3 Exponents

In this chapter, you learn about exponents and exponentiation.

Exponential Notation

An exponent is a small raised number written to the upper right of a quantity, which is called the base for the exponent. For example, consider the product 3 × 3 × 3 × 3 × 3, in which the same number is repeated as a factor multiple times. The exponential form for 3 × 3 × 3 × 3 × 3 is 3⁵. The number 3 is the base, and the small 5 to the upper right of 3 is the exponent.

The exponential form of a product of repeated factors is a shortened notation for the product.

Most commonly, you read 3⁵ as either “three to the fifth” or “three raised to the fifth power.” When the exponent is 1, as in 7¹, you say, “seven to the first.” When 2 is the exponent, as in 6², you say, “six squared,” and when 3 is the exponent, as in 5³, you say, “five cubed.”

Usually the exponent 1 is not written for expressions to the first power; that is, 7¹ = 7.

Problem

Express the exponential form in words.

a. 2⁵

b. 5⁶

c. 4³

d. (−13)²

e. 10¹

Solution

a. 2⁵

Step 1. Identify the base.

The base is 2.

Step 2. Identify the exponent.

The exponent is 5.

Step 3. Express 2⁵ in words.

2⁵ is “two to the fifth.”

b. 2⁶

Step 1. Identify the base.

The base is 5.

Step 2. Identify the exponent.

The exponent is 6.

Step 3. Express 5⁶ in words.

5⁶ is “five to the sixth.”

c. 4³

Step 1. Identify the base.

The base is 4.

Step 2. Identify the exponent.

The exponent is 3.

Step 3. Express 4³ in words.

4³ is “four cubed.”

d. (−13)²

Step 1. Identify the base.

The base is −13.

Step 2. Identify the exponent.

The exponent is 2.

Step 3. Express (−13)² in words.

(−13)² is “negative thirteen squared.”

When you use a negative number as the base in an exponential form, enclose the negative number in parentheses.

e. 10¹

Step 1. Identify the base.

The base is 10.

Step 2. Identify the exponent.

The exponent is 1.

Step 3. Express 10¹ in words.

10¹ is “ten to the first” or because 10¹ = 10, simply “10.”

Natural Number Exponents

When the exponent is a natural number, it tells you how many times to use the base as a factor.

Recall that the natural numbers are 1, 2, 3, and so on.

Problem

Write the indicated product in exponential form.

a. 2 × 2 × 2 × 2 × 2 × 2 × 2

b. (−3)(−3)(−3)(−3)(−3)(−3)

Solution

a. 2 × 2 × 2 × 2 × 2 × 2 × 2

Step 1. Count how many times 2 is a factor.

Step 2. Write the indicated product as an exponential expression with 2 as the base and 7 as the exponent.

2 × 2 × 2 × 2 × 2 × 2 × 2 = 2⁷

b. (−3)(−3)(−3)(−3)(−3)(−3)

Step 1. Count how many times −3 is a factor.

Step 2. Write the indicated product as an exponential expression with −3 as the base and 6 as the exponent.

(−3)(−3)(−3)(−3)(−3)(−3) = (−3)⁶

In the above problem, you must enclose the −3 in parentheses to show that −3 is the number that is used as a factor six times.

To evaluate the exponential form 3⁵, you do to the base what the exponent tells you to do. The result you get is the fifth power of 3, as shown in Figure 3.1.

Problem

Evaluate 3⁵.

Solution

Images

Step 1. Write 3⁵ in product form, using 3 as a factor five times.

3⁵ = 3 × 3 × 3 × 3 × 3

Step 2. Do the multiplication.

3⁵ = 3 × 3 × 3 × 3 × 3 = 243 (fifth power of 3)

3⁵ ≠ 3 × 5; 3⁵ = 243, but 3 × 5 = 15. Don’t multiply the base by the exponent! That is a common mistake.

Note: To express 3⁵ = 243 in words, say either “three to the fifth is two hundred forty-three” or “three raised to the fifth power is two hundred forty-three.”

Figure 3.1 Parts of an exponential form

Problem

Evaluate the expression.

a. 2⁵

b. 5⁶

c. 4³

d. (−13)²

e. 10¹

Solution

a. 2⁵

Step 1. Write 2⁵ in product form, using 2 as a factor five times.

2⁵ = 2 × 2 × 2 × 2 × 2

Step 2. Do the multiplication.

2 × 2 × 2 × 2 × 2 = 32

Step 3. Review the main results.

2⁵ = 2 × 2 × 2 × 2 × 2 = 32

b. 5⁶

Step 1. Write 5⁶ in product form, using 5 as a factor six times.

5⁶ = 5 × 5 × 5 × 5 × 5 × 5

Step 2. Do the multiplication.

5 × 5 × 5 × 5 × 5 × 5 = 15,625

Step 3. Review the main results.

5⁶ = 5 × 5 × 5 × 5 × 5 × 5 = 15,625

c. 4³

Step 1. Write 4³ in product form, using 4 as a factor three times.

4³ = 4 × 4 × 4

Step 2. Do the multiplication.

4 × 4 × 4 = 64

Step 3. Review the main results.

4³ = 4 × 4 × 4 = 64

d. (−13)²

Step 1. Write (−13)² in product form, using −13 as a factor two times.

(−13)=(−13)(−13)

Step 2. Do the multiplication.

(−13)(−13) = 169

Step 3. Review the main results.

(−13)² = (—13)(—13) = 169

e. 10¹

Step 1. Write 10 as a factor one time.

10¹ = 10

You likely are most familiar with natural number exponents, but natural numbers are not the only numbers you can use as exponents. Here are two other types of exponents.

Zero Exponents

A zero exponent on a nonzero number tells you to put 1 as the answer when you evaluate. Caution: It’s important to remember that when you use 0 as an exponent, the base cannot be 0. The expression 0⁰ has no meaning. You say, “zero to the zero power is undefined.”

Problem

Evaluate.

a. 2⁰

b. (−25)⁰

c. 0⁰

d. 100⁰

e. 1⁰

Solution

a. 2⁰

Step 1. The exponent is 0, so put 1 as the answer.

2⁰ = 1

b. (−25)°

Step 1. The exponent is 0, so put 1 as the answer.

(−25)° = 1

2⁰ ≠ 0 and also 2⁰ ≠ 2. A zero exponent gives 1 as the answer, so 2⁰ = 1.

c. 0°

Step 1. The exponent is 0, but 0⁰ has no meaning, so put undefined as the answer.

0⁰ is undefined.

d. 100⁰

Step 1. The exponent is 0, so put 1 as the answer. 100⁰ = 1

e. 1⁰

Step 1. The exponent is 0, so put 1 as the answer.

1⁰ = 1

Negative Exponents

A negative exponent on a nonzero number tells you to obtain the reciprocal of the corresponding expression that has a positive exponent. Caution: You cannot use 0 as a base for negative exponents. When you evaluate such expressions, you get 0 in the denominator, meaning that you have division by 0, which is undefined.

The reciprocal of a quantity is a fraction that has 1 in the numerator and the quantity in the denominator; for instance, 1 the reciprocal of 2⁵ is .