Exponentiation - Master High-Frequency Concepts and Skills for Algebra Proficiency—FAST - Easy Algebra Step-by-Step

Easy Algebra Step-by-Step: Master High-Frequency Concepts and Skills for Algebra Proficiency—FAST! (2012)

Chapter 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, xa is “x to the ath,” “x to the ath power,” or “x raised to the ath power.”

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


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


The result you get is the ath 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).

Image

Figure 4.1 Parts of an exponential form

Image Step 1. Write 35 in product form.

35 = 3 · 3 · 3 · 3 · 3

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.

Image Natural Number Exponents

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

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.

Image

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 cubeof 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! image


Problem Write the indicated product as an exponential expression.

image

Solution

image

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

Image

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

image

Image Step 1. Count how many times –3 is a factor.

Image

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

image

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.


image


Problem Evaluate.

a. 25

b. (–2)5

c. (0.6)2

d. Image

e. 0100

f. 13

g. (1 + 1)3

Solution

a. 25

Image 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

Image Step 1. Write (–2)5 in product form.

image

Step 2. Do the multiplication.

image

Image 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. Image

Image Step 1. Write Image in product form.

Image

Step 2. Do the multiplication.

Image

e. 0100

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

0100 = 0

f. 13

Image Step 1. Write 13 in product form.

13 = 1 · 1 · 1

Step 2. Do the multiplication.

image

Image 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.)

image


image


Step 2. Write 23 in product form.

23 = 2 · 2 · 2

Step 3. Do the multiplication.

23 = 2 · 2 · 2 = 8

Zero and Negative Integer Exponents

Image Zero Exponent

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


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


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


image


Problem Evaluate.

a. (–2)0

b. (0.6)0

c. Image

d. π0

e. 10

Solution

a. (–2)0

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

(–2)0 = 1

b. (0.6)0

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

(0.6)0 = 1

c. Image

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

Image

d. π0

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

π0 = 1

e. 10

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

10 = 1

Image Negative Integer Exponents

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

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


Image. A negative exponent does not make a power negative.


Problem Evaluate.

image

d. Image

Solution

a. 2–5

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

Image

Step 2. Evaluate 25.

Image

image


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


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

Image

Step 2. Evaluate (–2)5.

Image


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.


c. (0.6)–2

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

Image

Step 2. Evaluate (0.6)2.

Image

d. Image

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

Image

Step 2. Evaluate Image and simplify.

Image

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

Image; thus, Image. Apply this rule in the following problem.

Problem Simplify.

a. Image

b. Image

c. Image

Solution

a. Image

Image Step 1. Apply Image.

Image


Image. Keep the same base for the corresponding positive exponent version.


Step 2. Evaluate 25.

Image

b. Image

Image Step 1. Apply Image.

Image

Step 2. Evaluate (–2)5.

Image

c. Image

Image Step 1. Apply Image.

Image

Step 2. Evaluate (0.6)2.

Image

Unit Fraction and Rational Exponents

Image Unit Fraction Exponents

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

Solution

a. (–27)1/3

Image Step 1. Apply Image.

Image

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

Image

b. (0.25)1/2

Image Step 1. Apply Image.

Image

Step 2. Find the principal square root of 0.25.

Image

c. (–16)1/4

Image Step 1. Apply Image.

Image

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

Image is not defined for real numbers.


(–16)1/4 ≠ –2. –2 · –2 · –2 · –2 = 16, not –16.


d. Image

Image Step 1. Apply Image.

Image

Step 2. Find the principal fifth root of Image.

Image


When you evaluate exponential expressions that have unit fraction exponents, you should practice doing Step 1 mentally. For instance, 491/2 = 7, (–8)1/3 = –2, Image, and so forth.


Image Rational Exponents

If x is a real number and m and n are natural numbers, then (a) xm/n = (x1/n)m or (b) xm/n = (xm)1/n, provided that in all cases even roots of negative numbers do not occur.

When you evaluate the exponential expression xm/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 xm/n = (x1/n)m.

a. Image

b. (36)3/2

Solution

a. Image

Image Step 1. Rewrite Image using xm/n = (x1/n)m.

Image

Step 2. Find Image.

Image

Step 3. Raise Image to the second power.

Image

Step 4. Review the main results.

Image

b. (36)3/2

Image Step 1. Rewrite (36)3/2 using xm/n = (x1/n)m.

(36)3/2 = (361/2)3

Step 2. Find (36)1/2.

(36)1/2 = 6

Step 3. Raise 6 to the third power.

63 = 216

Step 4. Review the main results.

(36)3/2 = (361/2)3 = 63 = 216


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


Problem Evaluate using xm/n = (xm)1/n.

a. Image

b. (36)3/2

Solution

a. Image

Image Step 1. Rewrite Image using xm/n = (xm)1/n.

Image

Step 2. Find Image.

Image

Step 3. Find Image.

Image

Step 4. Review the main results.

Image

b. (36)3/2

Image Step 1. Rewrite (36)3/2 using xm/n = (xm)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 the main results.

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

Image 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. Image

6. 09

7. (1 + 1)5

8. (–2)0

9. 3–4

10. (–4)–2

11. (0.3)–2

12. Image

13. (–125)1/3

14. (0.16)1/2

15. (–121)1/4

16. Image

17. (–27)2/3

18. Image

For 19 and 20, simplify.

19. Image

20. Image