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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

You need to recall this terminology while you work through this lesson:

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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:
  1. Add the exponents
  2. Use the common base

Examples: Multiplying Terms with Exponents

More Examples: Multiplying Terms with Exponents

  1. 32 x 31 = 33
  2. 54 x 50 = 54 = 1
  3. 96 x 92 = 9(62) = 9 4
  4. 103 x 108 = 10(38) = 105
  5. 105 x 105 = 10(5 + 5) = 100 = 1 (any term to the zero power is equal to 1)

 

Examples & Exercises: Multiplying Terms with Exponents

Use a calculator to find the square roots of the given numbers. Round your answer to the nearest hundredth.

Work these problems until you can do the work without making any errors.

Dividing  Terms with Exponents

Procedure

To divide powers that have the same base:
  1. Subtract the exponents (divisor from dividend)
  2. 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

More Examples: Dividing Terms with Exponents

  1. 32 31 = 3(21) =  31 = 3 (any number to the power of 1 is that number)

  2. 54 50 = 5(40)  = 54

  3. 96 92 = 9(6+2) = 98

  4. 103 108 = 10[3(8)] = 1011

  5. 105 105 = 10(5 5) = 10-10

Examples & 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 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:

  1. Evaluate each term with normal decimal notation.
  2. Complete the multiplication or division.

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:

  1. Rewrite each number with normal decimal notation.
  2. Complete the addition or subtraction.

Examples

23 + 33 = 8 + 27 =  35

42 23 = 16 8 = 8

David L. Heiserman, Editor

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All Rights Reserved

Revised: June 06, 2015