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).
Figure 4.1 Parts of an exponential form
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.
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 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!
Problem Write the indicated product as an exponential expression.
Solution
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.
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.
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.
Problem Evaluate.
a. 25
b. (–2)5
c. (0.6)2
d.
e. 0100
f. 13
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.
Step 2. Do the multiplication.
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 1. 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. 13
Step 1. Write 13 in product form.
13 = 1 · 1 · 1
Step 2. Do the multiplication.
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.)
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
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.
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.
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; .
Step 1. Write the reciprocal of the corresponding positive exponent version of (–2)–5.
Step 2. Evaluate (–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.
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
a.
Step 1. Apply .
. Keep the same base for the corresponding positive exponent version.
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, 491/2 = 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) 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.
b. (36)3/2
Solution
a.
Step 1. Rewrite using xm/n = (x1/n)m.
Step 2. Find .
Step 3. Raise to the second power.
Step 4. Review the main results.
b. (36)3/2
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
, but . Don’t multiply the base by the exponent!
Problem Evaluate using xm/n = (xm)1/n.
a.
b. (36)3/2
Solution
a.
Step 1. Rewrite using xm/n = (xm)1/n.
Step 2. Find .
Step 3. Find .
Step 4. Review the main results.
b. (36)3/2
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
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. 09
7. (1 + 1)5
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.