Exponents, Polynomials, and Polynomial Functions. Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 5.1, 1

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1 5 Exponents, Polynomials, and Polynomial Functions Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 5.1, 1

2 5.1 Integer Exponents R.1 Fractions and Scientific Notation Objectives 1. Use the product rule for exponents. 2. Define 0 and negative exponents.. Use the quotient rule for exponents. 4. Use the power rule for exponents. 5. Simplify exponential expressions. 6. Use the rules for exponents with scientific notation. Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 5.1, 2

3 Use the product rule for exponents. The products of exponential expressions with the same base are found by adding exponents. Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 5.1,

4 Example 1 Using 0 as an Exponent Apply the product rule for exponents, if possible, in each case. 2 a b. x x x 2 = = x x x 5 = 5 = = 1 6 x ++ x 10 c. ( 2p 4 )( p)( 4p ) 4 1 = 2( )(4)p pp 4 1 = 24p ++ 8 = 24p d. 7a b 2 The bases are not the same, so the product rule does not apply. Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 5.1, 4

5 Define 0 and negative exponents Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 5.1, 5

6 Example 2 Evaluate. Using 0 as an Exponent a. b. 0 5 = 1 0 ( 7) = 1 c = (7 ) = 1 0 d. ( 7) = 1 e. f = 1+ 1= 2 0 (8 y ) = 1 Any nonzero quantity raised to the zero power equals 1. Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 5.1, 6

7 Negative Exponents Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 5.1, 7

8 Example Using Negative Exponents Write with only positive exponents. a. 1 = b p 1 2 (2 p) c. ( ) z = 6 z = z = d. = 1 7 e. 1 a 1 ( a) = ( a ) a 4 4 = f. 4 4 Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 5.1, 8

9 Special Rules for Negative Exponents Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 5.1, 9

10 Use the quotient rule for exponents. Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 5.1, 10

11 Example 5 Using the Quotient Rule for Exponents Apply the quotient rule for exponents, if possible, write each result using only positive exponents. 7 5 a. 5 7 = 5 4 = 5 b. w w 7 ( 7) = w 7 = w + = w 4 c. a b 4 The quotient rule does not apply because the bases are different. Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 5.1, 11

12 Use the power rule for exponents. Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 5.1, 12

13 Example 6 Use the power rule for exponents. Simplify, using the power rules. Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 5.1, 1

14 More Special Rules for Negative Exponents Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 5.1, 14

15 Example 7 Using Negative Exponents with Fractions Write each expression with positive exponents and then evaluate. a. 2 = 2 = 2 = 27 8 b = = 49 9 Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 5.1, 15

16 Using the Definitions and Rules for Exponents Example 8 Simplify each expression so that no negative exponents appear in the final result. Assume that all variables represent nonzero real numbers. a. 2 4 ( 4) 2 x x x = x + + = = x 5 1 x 5 c. xy z k k = 1 2 xy z 9k y = 2 4 xz 6 2 b. x z 2 y = yz x 2 Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 5.1, 16

17 Scientific Notation Scientists often deal with extremely large and extremely small numbers. For example: The distance from the sun to the Earth is approximately 150,000,000 kilometers. X-ray The wavelength of x-rays is meter. Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 5.1, 17

18 Scientific Notation It is often simpler to write these very large or very small numbers using scientific notation. Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 5.1, 18

19 Converting to Scientific Notation To write numbers in scientific notation, we use the following steps. If the number is negative, use the steps above and then label your results as a negative number. Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 5.1, 19

20 Example 9 Writing Numbers in Scientific Notation Write each number in scientific notation. a. 150, Move the decimal point 8 places to the right b Move the decimal point 6 places to the left Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 5.1, 20

21 Converting from Scientific Notation to Standard Notation Example 10 Write each number in standard notation. 7 a Move the decimal point 7 places to the right. 69,00,000 b Move the decimal point 6 places to the left Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 5.1, 21

22 Using Scientific Notation to Solve Problems Example 12 In 1990, the national health care expenditure in the United States was $714.0 billion. By 2009, this figure had risen by a factor of.5 that is, it more than tripled in about 20 yr. a. Write the 1990 health care expenditure using scientific notation billion 9 = = ( ) = Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 5.1, 22

23 Using Scientific Notation to Solve Problems Example 12 In 1990, the national health care expenditure in the United States was $714.0 billion. By 2009, this figure had risen by a factor of.5 that is, it more than tripled in about 20 yr. b. What was the expenditure in 2009? Multiply the result in part (a) by = ( ).5 11 = ( ) = = ( ) = Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 5.1, 2