## 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 3^{5}. 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 3^{5} is read as “three to the fifth.” Other ways you might read 3^{5} are “three to the fifth power” or “three raised to the fifth power.” In general, *x ^{a}* is “

*x*to the

*a*th,” “

*x*to the

*a*th power,” or “

*x*raised to the

*a*th power.”

*Exponentiation* is the act of evaluating an exponential expression, *x ^{a}*.

*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 *a*th *power* of the base. For instance, to evaluate 3^{5}, 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 3^{5} in product form.

3^{5} = 3 · 3 · 3 · 3 · 3

*Step 2*. Do the multiplication.

3^{5} = 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, 5^{4} 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, 5^{1}, you usually omit the exponent and simply write 5. The second power of a number is the *square* of the number; read 5^{2} as “five squared.” The third power of a number is the *cube*of the number; read 5^{3} as “five cubed.” Beyond the third power, read 5^{4} as “five to the fourth,” read 5^{5} as “five to the fifth,” read 5^{6} 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**. 2^{5}

**b**. (–2)^{5}

**c**. (0.6)^{2}

**d**.

**e**. 0^{100}

**f**. 1^{3}

**g**. (1 + 1)^{3}

**Solution**

**a**. 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

**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**. 0^{100}

*Step 1*. Because 0^{100} has 0 as a factor 100 times, the product is 0.

0^{100} = 0

**f**. 1^{3}

*Step 1*. Write 1^{3} in product form.

1^{3} = 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 2^{3} in product form.

2^{3} = 2 · 2 · 2

*Step 3*. Do the multiplication.

2^{3} = 2 · 2 · 2 = 8

**Zero and Negative Integer Exponents**

** Zero Exponent**

If *x* is a nonzero real number, then *x*^{0} = 1.

0^{0} is undefined; it has no meaning. 0^{0} ≠ 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**. *π*^{0}

**e**. 1^{0}

**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**. *π*^{0}

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

*π*^{0} = 1

**e**. 1^{0}

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

1^{0} = 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 2^{5}.

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 2^{5}.

**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 *n*th 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})

*or (b)*

^{m}*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

*n*th root of

*x*first and then raise the result to the

*m*th power, or you can raise

*x*to the

*m*th power first and then find the

*n*th 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 the 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.

6^{3} = 216

*Step 4*. Review the main results.

(**36**)^{3/2} = (**36 ^{1/2}**)

^{3}= 6

^{3}= 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 the main results.

**b**. (**36**)^{3/2}

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

*x*)

^{m}^{1/n}.

(36)^{3/2} = (36^{3})^{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} = (36^{3})^{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__. 0^{9}

__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__.