Chapter 5—Powers, Exponents, and Roots
5-3 Working with Exponents
When you complete the work for this section, you should be able to: - State the rules for multiplying and dividing terms with exponents
- Demonstrate how to multiply and divide values expressed with exponents
- Describe the procedure for adding and subtracting terms with exponents
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You need to recall this terminology while you work through this lesson:
Multiplying and Dividing Terms with Exponents
| Rule Numbers with exponents can be directly multiplied or divided only when they have the same base. |
The following expressions can be directly multiplied or divided because they have the same base:
23 x 22 = _____ 84 ÷ 82 = _____ 4.4-2 x 4.42 + _____
These expressions cannot be directly multiplied or divided because they do not have the same base.
32 x 42 = _____ 4.86 ÷ 2.45 = _____ 2.3-2 x 23-2 = _____
Multiplying Terms with Exponents
| Procedure To multiply powers that have the same base: - Add the exponents
- Use the common base
|  | |
Examples: Multiplying Terms with Exponents
Problem 1 Multiply 23 x 22 | |
| Procedure | |
- Add the exponents
| 23 x 22
3 + 2 = 5 |
- Determine the common base
| 23 x 22
The common base is 2 |
- Complete the procedure
| - 25
|
Solution 23 x 22 = 25 | |
Problem 2 Multiply 4.22 x 4.24 | |
| Procedure | |
- Add the exponents
| 4.22 x 4.24
2 + 4 = 6 |
- Determine the common base
| 4.22 x 4.24
The common base is 4.2 |
- Complete the procedure
| - 4.26
|
Solution 4.22 x 4.24 = 4.22 + 4 = 4.26 | |
More Examples: Multiplying Terms with Exponents
- 32 x 31 = 33
- 54 x 50 = 54 = 1
- 96 x 9–2 = 9(6–2) = 9 4
- 103 x 10–8 = 10(3–8) = 10–5
- 10–5 x 105 = 10(–5 + 5) = 100 = 1 (any term to the zero power is equal to 1)
Examples and Exercises: Multiplying Terms with Exponents
| Use these exercises to test your understanding and build your skill level. Continue working them until you no longer make errors. | |
Dividing Terms with Exponents
| Procedure To divide powers that have the same base: - Subtract the exponents (divisor from dividend)
- Use the common base
Note: Subtract the exponent of the divisor from the exponent of the dividend. 
If the expression is shown as a fraction, subtract the exponent of he denominator from the exponent of the numerator. 
|  | |
Examples: Dividing Terms with Exponents
Problem 1 Divide 85 ÷ 82 | |
| Procedure | |
- Subtract the exponents
| 85 ÷ 82
5 – 2 = 3 |
- Determine the common base
| 85 ÷ 82
The common base is 8 |
- Complete the procedure
| - 83
|
Solution 85 ÷ 82 = 83 | |
Problem 2 Divide 1214 ÷ 1210 | |
| Procedure | |
- Subtract the exponents
| 1214 ÷ 1210
14 – 10 = 4 |
- Determine the common base
| 1214 ÷ 1210
The common base is 12 |
- Complete the procedure
| - 124
|
Solution 214 ÷ 1210 =124 | |
Problem 3 Divide  | |
| Procedure | |
- Subtract the exponents
| 
8 – 3 = 5
|
- Determine the common base
| 
The common base is 6 |
- Complete the procedure
| - 65
|
Solution  | |
More Examples: Dividing Terms with Exponents
- 32 ÷ 31 = 3(2–1) = 31 = 3 (any number to the power of 1 is that number)
- 54 ÷ 50 = 5(4–0) = 54
- 96 ÷ 9–2 = 9(6+2) = 98
- 103 ÷ 10–8 = 10[3–(–8)] = 1011
- 10–5 ÷ 105 = 10(–5 – 5) = 10-10
Examples and Exercises: Dividing Terms with Exponents
| Use these exercises to test your understanding and build your skill level. Continue working them until you no longer make errors. | |
Working with Exponential Terms that Do Not Have a Common Base
Consider these examples of multiplication and division of terms that have exponents.
123 x 103 = ____
48 ÷ 62 = _____
When the exponent terms do not have a common base, you simply have to rewrite the terms in normal decimal form and complete the multiplication/division in that form.
| Procedure To multiply or divide exponent terms that do not have the same base: - Evaluate each term with normal decimal notation.
- Complete the multiplication or division.
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Examples
23 x 33 = 8 x 27 = 216
42 ÷ 23 = 16 ÷ 8 = 2
Adding and Subtracting Terms with Exponents
There are no special rules for adding and subtracting numbers that are written with exponents. Each number must first be converted to its ordinary decimal form, then complete the addition/subtraction operation.
| Procedure To add or subtract numbers written with exponents: - Rewrite each number with normal decimal notation.
- Complete the multiplication or division.
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Examples
23 + 33 = 8 + 27 = 35
42 – 23 = 16 – 8 = 8
Examples & Exercises
