4

Exponentiation

This chapter presents a detailed discussion of exponents. Working efficiently and accurately with exponents will serve you well in algebra.

Exponents

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 shortened notation for 3 · 3 · 3 · 3 · 3 is 3⁵. This representation of the product is an exponential expression. The number 3 is the base, and the small 5 to the upper right of 3 is the exponent. Most commonly, the exponential expression 3⁵ is read as “three to the fifth.” Other ways you might read 3⁵ are “three to the fifth power” or “three raised to the fifth power.” In general, xⁿ is “x to the nth,” “x to the nth power,” or “x raised to the nth power.”

Exponentiation is a big word, but it just means that you do to the base what the exponent tells you to do to it.

Exponentiation is the act of evaluating an exponential expression, xⁿ.

The result you get is the nth power of the base. For instance, to evaluate 3⁵, which has the natural number 5 as an exponent, perform the multiplication as shown here (see Figure 4.1).

Step 1. Write 3⁵ in product form.

3⁵ = 3 · 3 · 3 · 3 · 3

Figure 4.1 Parts of an exponential form

Step 2.   Do the multiplication.

3⁵ = 3 · 3 · 3 · 3 · 3 = 243 (the fifth power of 3)

The following discussion tells you about the different types of exponents and what they tell you to do to the base.

Natural Number Exponents

You likely are most familiar with natural number exponents.

Natural Number Exponents

If x is a real number and n is a natural number, then

For instance, 5⁴ has a natural number exponent, namely, 4. The exponent 4 tells you how many times to use the base 5 as a factor. When you do the exponentiation, the product is the fourth power of 5, as shown in Figure 4.2.

Figure 4.2 Fourth power of 5

For the first power of a number, for instance, 5¹, you usually omit the exponent and simply write 5. The second power of a number is the square of the number; read 5² as “five squared.” The third power of a number is the cube of the number; read 5³ as “five cubed.” Beyond the third power, read 5⁴ as “five to the fourth,” read 5⁵ as “five to the fifth,” read 5⁶ as “five to the sixth,” and so on.

Don’t multiply the base by the exponent! 5² ≠ 5 · 2 = 10, 5³ ≠ 5 · 3 = 15, 5⁴ ≠ 5 · 4 = 20, etc. xⁿ ≠ x · n.

Problem   Write the indicated product as an exponential expression.

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. Only the 3 will be raised to the power unless parentheses are used to indicate otherwise.

(−3)⁶ ≠ −3⁶.(−3)⁶ = 729, but −3⁶ = −729.

Problem   Evaluate.

a. 2⁵

b. (−2)⁵

c. (0.6)²

d.

e. 0¹⁰⁰

f. −5²

g. (1 + 1)³

Solution

a. 2⁵

Step 1.   Write 2⁵ in product form.

2⁵ = 2 · 2 · 2 · 2 · 2

Step 2.   Do the multiplication.

2⁵ = 2 · 2 · 2 · 2 · 2 = 32

b. (−2)⁵

Step 1.   Write (−2)⁵ in product form.

(−2)⁵ = −2 · −2 · −2 · −2 · −2

Step 2.   Do the multiplication.

(−2)⁵ = −2 · −2 · −2 · −2 · −2 = −32

c. (0.6)²

Step 1.   Write (0.6)² in product form.

(0.6)² = (0.6)(0.6)

Step 2.   Do the multiplication.

(0.6)² = (0.6)(0.6) = 0.³⁶

d.

Step 3.   Write in product form.

Step 2.   Do the multiplication.

e. 0¹⁰⁰

Step 1.   Because 0¹⁰⁰ has 0 as a factor 100 times, the product is 0.

0¹⁰⁰ = 0

f. −5²

Step 1.   Keep the − symbol in front and write 5² in product form.

−5² = −5 · 5

Step 2.   Do the multiplication.

−5² = −5 · 5 = −25

An exponent applies only to the number or grouped quantity to which it is attached. Thus, −5² = −25, not 25.

g. (1 + 1)³

Step 1.   Add 1 and 1 because you want to cube the quantity 1 + 1. (See Chapter 5 for a discussion of parentheses as a grouping symbol.)

(1 + 1)³ = 2³

Step 2.   Write 2³ in product form.

2³ = 2 · 2 · 2

Step 3.   Do the multiplication.

2³ = 2 · 2 · 2 = 8

(1 + 1)³ ≠ 1³ + 1³ = (1 + 1)³ = 2³ = 8, but 1³ + 1³ = 1 + 1 = 2.

Zero and Negative Integer Exponents

0⁰ is undefined; it has no meaning. 0⁰ ≠ 0.

x⁰ ≠ 0; x⁰ = 1, provided x ≠ 0.

Zero Exponent

If x is a nonzero real number, then x⁰ = 1.

A zero exponent on a nonzero number tells you to put 1 as the answer when you evaluate.

Problem   Evaluate.

a. (−2)⁰

b. (0.6)⁰

c.

d. π⁰

e. 1⁰

Solution

a. (−2)⁰

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

(−2)⁰ = 1

b. (0.6)⁰

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

(0.6)⁰ = 1

c.

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

d. π⁰

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

π⁰ = 1

e. 1⁰

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

1⁰ = 1

Negative Integer Exponents

If x is a nonzero real number and n is a natural number, then .

A negative integer exponent on a nonzero number tells you to obtain the reciprocal of the corresponding exponential expression that has a positive exponent.

A negative exponent does not make a power negative.

Problem   Evaluate.

a. 2⁻⁵

b. (−2)⁻⁵

c. (0.6)⁻²

d.

Solution

a. 2⁻⁵

Step 1.   Write the reciprocal of the corresponding positive exponent version of 2⁻⁵.

Step 2.   Evaluate 2⁵.

As you can see, the negative exponent did not make the answer negative;

b. (−2)⁻⁵

Step 1.   Write the reciprocal of the corresponding positive exponent version of (−2)⁻⁵.

When you evaluate (−2)⁻⁵, the answer is negative because (−2)⁵ is negative. The negative exponent is not the reason (−2)⁵ is negative.

Step 2.   Evaluate (−2)⁵.

c. (0.6)⁻²

Step 1.   Write the reciprocal of the corresponding positive exponent version of (0.6)⁻².

Step 2.   Evaluate (0.6)².

d.

Step 1.   Write the reciprocal of the corresponding positive exponent version of .

Step 2.   Evaluate and simplify.

Notice that because , the expression can be simplified as follows:

thus, . Apply this rule in the following problem.

Problem   Simplify.

a.

b.

c.

Solution

Keep the same base for the corresponding positive exponent version.

a.

Step 1.   Apply

Step 2.   Evaluate 2⁵.

b.

Step 1.   Apply

Step 2.   Evaluate (−2)⁵.

c.

Step 1.   Apply

Step 2.   Evaluate (0.6)².

Unit Fraction and Rational Exponents

Unit Fraction Exponents

If x is a real number and n is a natural number, then , provided that, when n is even, x ≥ 0.

A unit fraction exponent on a number tells you to find the principal nth root of the number.

Problem   Evaluate, if possible.

a. (−27)^1/3

b. (0.25)^1/2

c. (−16)^1/4

d.

Solution

a. (−27)^1/3

Step 1.   Apply

Step 2.   Find the principal cube root of −27.

b. (0.25)^1/2

Step 1.   Apply

Step 2.   Find the principal square root of 0.25.

c. (−16)^1/4

Step 1.   Apply

Step 2.   −16 is negative and 4 is even, so (−16)^1/4 is not a real number.

is not defined for real numbers.

(−16)^1/4 ≠ −2.

−2 ·−2 ·−2 ·−2 =16, not −16.

d.

Step 1.   Apply

Step 2.   Find the principal fifth root of

When you evaluate exponential expressions that have unit fraction exponents, you should practice doing Step 1 mentally. For instance, 49^1/2 = 7, (−8)^1/3 = −2, , and so forth.

Rational Exponents

If x is a real number and m and n are natural numbers, then (a) x^m/n = (x^1/n)^m or (b) x^m/n = (x^m)^1/n, provided that in all cases even roots of negative numbers do not occur.

When you evaluate the exponential expression x^m/n, you can find the nth root of x first and then raise the result to the mth power, or you can raise x to the mth power first and then find the nth root of the result. For most numerical situations, you usually will find it easier to find the root first and then raise to the power (as you will observe from the sample problems shown here).

Problem   Evaluate using x^m/n = (x^1/n)^m.

a.

b. (36)^3/2

Solution

a.

Step 1.   Rewrite using x^m/n = (x^1/n)^m.

Step 2.   Find .

Step 3.   Raise to the second power.

Step 4.   Review your main results.

b. (36)^3/2

Step 1.   Rewrite (36)^3/2 using x^m/n = (x^1/n)^m.

(36)^3/2 = (36^1/2)³

Step 2.   Find (36)^1/2.

(36)^1/2 = 6

Step 3.   Raise 6 to the third power.

6³ = 216

Step 4.   Review your main results.

(36)^3/2 = (36^1/2)³ = 6³ = 216

, but . Don’t multiply the base by the exponent!

Problem   Evaluate using x^m/n = (x^m)^1/n.

a.

b. (36)^3/2

Solution

a.

Step 1.   Rewrite using x^m/n = (x^m)^1/n.

Step 2.   Find

Step 3.   Find

Step 4.   Review your main results.

b. (36)^3/2

Step 1.   Rewrite (36)^3/2 using x^m/n = (x^m)^1/n.

(36)^3/2 = (36³)^1/2

Step 2.   Find (36)³.

(36)³ = 46,656

Step 3.   Find (46,656)^1/2.

(46,656)^1/2 = 216

Step 4.   Review your main results.

(36)^3/2 = (36³)^1/2 = (46,656)^1/2 = 216

Exercise 4

For 1 and 2, write the indicated product as an exponential expression.

1. −4 ·−4 ·−4 ·−4 ·−4

2. 8 · 8 · 8 · 8 · 8 · 8 · 8

For 3–18, evaluate, if possible.

3. (−2)⁷

4. (0.3)⁴

5.

6. −2⁴

7. (1 + 1)⁴

8. (−2)⁰

9. 3⁻⁴

10. (−4)⁻²

11. (0.3)⁻²

12.

13. (−125)^1/3

14. (0.16)^1/2

15. (−121)^1/4

16.

17. (−27)^2/3

18.

For 19 and 20, simplify.

19.

20.