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Chapter 5—Powers, Exponents, and Roots 53 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: 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: 2^{3} x 2^{2} = _____ 8^{4} ÷ 8^{2} = _____ 4.4^{2} x 4.4^{2} + _____ These expressions cannot be directly multiplied or divided because they do not have the same base. 3^{2} x 4^{2} = _____ 4.8^{6} ÷ 2.4^{5} = _____ 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 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 & 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:  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 More Examples: Dividing Terms with Exponents 
3^{2} ÷ 3^{1} = 3^{(2–1) }= 3^{1} = 3 (any number to the power of 1 is that number) 
5^{4} ÷ 5^{0} = 5^{(4–0) } = 5^{4} 
9^{6} ÷ 9^{–2} = 9^{(6+2)} = 9^{8} 
10^{3} ÷ 10^{–8} = 10^{[3–(–8)]} = 10^{11} 
10^{–}^{5} ÷ 10^{5} = 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. 12^{3} x 10^{3} = ____ 4^{8} ÷ 6^{2} = _____ 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:  Evaluate each term with normal decimal notation.
 Complete the multiplication or division.
 Examples 2^{3} x 3^{3} = 8 x 27 = 216 4^{2} ÷ 2^{3} = 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 addition or subtraction.
 Examples 2^{3} + 3^{3} = 8 + 27 = 35 4^{2} – 2^{3} = 16 – 8 = 8
