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Section 1Chapter 5

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1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objectives 2 6 5 3 4 Integer Exponents and Scientific Notation Use the product rule for exponents. Define 0 and negative exponents. Use the quotient rule for exponents. Use the power rules for exponents. Simplify exponential expressions. Use the rules for exponents with scientific notation. 5.1

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. We use exponents to write products of repeated factors. For example, 2 5 is defined as 2 2 2 2 2 = 32. The number 5, the exponent, shows that the base 2 appears as a factor five times. The quantity 2 5 is called an exponential or a power. We read 2 5 as “2 to the fifth power” or “2 to the fifth.” Integer Exponents and Scientific Notation Slide 5.1- 3

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Use the product rule for exponents. Objective 1 Slide 5.1- 4

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Product Rule for Exponents If m and n are natural numbers and a is any real number, then a m a n = a m + n. That is, when multiplying powers of like bases, keep the same base and add the exponents. Slide 5.1- 5 Use the product rule for exponents. Be careful not to multiply the bases. Keep the same base and add the exponents.

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Apply the product rule, if possible, in each case. m 8 m 6 m 5 p 4 (–5p 4 ) (–9p 5 ) (–3x 2 y 3 ) (7xy 4 ) = m 8+6 = m 14 Cannot be simplified further because the bases m and p are not the same. The product rule does not apply. = 45p 9 = (–5)(–9)(p 4 p 5 )= 45p 4+5 = –21x 3 y 7 = (–3)(7) x 2 xy 3 y 4 = –21x 2+1 y 3+4 Slide 5.1- 6 CLASSROOM EXAMPLE 1 Using the Product Rule for Exponents Solution:

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Define 0 and negative exponents. Objective 2 Slide 5.1- 7

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Zero Exponent If a is any nonzero real number, then a 0 = 1. Slide 5.1- 8 Define 0 and negative exponents. The expression 0 0 is undefined.

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Evaluate. 29 0 (–29) 0 –29 0 8 0 – 15 0 = 1 = – (29 0 ) = –1 = 1 – 1 = 0 Slide 5.1- 9 CLASSROOM EXAMPLE 2 Using 0 as an Exponent Solution:

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Negative Exponent For any natural number n and any nonzero real number a, A negative exponent does not indicate a negative number; negative exponents lead to reciprocals. Slide 5.1- 10 Define 0 and negative exponents.

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Write with only positive exponents. 6 -5 (2x) -4, x ≠ 0 –7p -4, p ≠ 0 Evaluate 4 -1 – 2 -1. Slide 5.1- 11 CLASSROOM EXAMPLE 3 Using Negative Exponents Solution:

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Evaluate. Slide 5.1- 12 CLASSROOM EXAMPLE 4 Using Negative Exponents Solution:

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Special Rules for Negative Exponents If a ≠ 0 and b ≠ 0, then and Slide 5.1- 13 Define 0 and negative exponents.

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Use the quotient rule for exponents. Objective 3 Slide 5.1- 14

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Quotient Rule for Exponents If a is any nonzero real number and m and n are integers, then That is, when dividing powers of like bases, keep the same base and subtract the exponent of the denominator from the exponent of the numerator. Slide 5.1- 15 Use the quotient rule for exponents. Be careful when working with quotients that involve negative exponents in the denominator. Write the numerator exponent, then a subtraction symbol, and then the denominator exponent. Use parentheses.

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Apply the quotient rule, if possible, and write each result with only positive exponents. Cannot be simplified because the bases x and y are different. The quotient rule does not apply. Slide 5.1- 16 CLASSROOM EXAMPLE 5 Using the Quotient Rule for Exponents Solution:

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Use the power rules for exponents. Objective 4 Slide 5.1- 17

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Power Rule for Exponents If a and b are real numbers and m and n are integers, then and That is, a) To raise a power to a power, multiply exponents. b) To raise a product to a power, raise each factor to that power. c) To raise a quotient to a power, raise the numerator and the denominator to that power. Slide 5.1- 18 Use the power rules for exponents.

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Simplify, using the power rules. Slide 5.1- 19 CLASSROOM EXAMPLE 6 Using the Power Rules for Exponents Solution:

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Special Rules for Negative Exponents, Continued If a ≠ 0 and b ≠ 0 and n is an integer, then and That is, any nonzero number raised to the negative nth power is equal to the reciprocal of that number raised to the nth power. Slide 5.1- 20 Use the power rules for exponents.

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Write with only positive exponents and then evaluate. Slide 5.1- 21 CLASSROOM EXAMPLE 7 Using Negative Exponents with Fractions Solution:

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Definition and Rules for Exponents For all integers m and n and all real numbers a and b, the following rules apply. Product Rule Quotient Rule Zero Exponent Slide 5.1- 22 Use the power rules for exponents.

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Definition and Rules for Exponents, Continued Negative Exponent Power Rules Special Rules Slide 5.1- 23 Use the power rules for exponents.

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Simplify exponential expressions. Objective 5 Slide 5.1- 24

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Simplify. Assume that all variables represent nonzero real numbers. Slide 5.1- 25 CLASSROOM EXAMPLE 8 Using the Definitions and Rules for Exponents Solution:

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Combination of rules Slide 5.1- 26 CLASSROOM EXAMPLE 8 Using the Definitions and Rules for Exponents (cont’d) Simplify. Assume that all variables represent nonzero real numbers. Solution:

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Use the rules for exponents with scientific notation. Objective 6 Slide 5.1- 27

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. In scientific notation, a number is written with the decimal point after the first nonzero digit and multiplied by a power of 10. This is often a simpler way to express very large or very small numbers. Use the rules for exponents with scientific notation. Slide 5.1- 28

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Scientific Notation A number is written in scientific notation when it is expressed in the form a × 10 n where 1 ≤ |a| < 10 and n is an integer. Slide 5.1- 29 Use the rules for exponents with scientific notation.

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Converting to Scientific Notation Step 1 Position the decimal point. Place a caret, ^, to the right of the first nonzero digit, where the decimal point will be placed. Step 2 Determine the numeral for the exponent. Count the number of digits from the decimal point to the caret. This number gives the absolute value of the exponent on 10. Step 3 Determine the sign for the exponent. Decide whether multiplying by 10 n should make the result of Step 1 greater or less. The exponent should be positive to make the result greater; it should be negative to make the result less. Slide 5.1- 30 Use the rules for exponents with scientific notation.

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Write the number in scientific notation. Step 1 Place a caret to the right of the 2 (the first nonzero digit) to mark the new location of the decimal point. Step 2 Count from the decimal point 7 places, which is understood to be after the caret. Step 3 Since 2.98 is to be made greater, the exponent on 10 is positive. 29,800,000 29,800,000 = 2.9,800,000. Decimal point moves 7 places to the left 29,800,000 = 2.98 × 10 7 Slide 5.1- 31 CLASSROOM EXAMPLE 9 Writing Numbers in Scientific Notation Solution: ^

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Step 3 Since 5.03 is to be made less, the exponent 10 is negative. 0.0000000503 0.0000000503 = 0.00000005.03 0.0000000503 = 5.03 × 10 −8 Step 1 Place a caret to the right of the 5 (the first nonzero digit) to mark the new location of the decimal point. Step 2 Count from the decimal point 8 places, which is understood to be after the caret. Writing Numbers in Scientific Notation (cont’d) Slide 5.1- 32 CLASSROOM EXAMPLE 9 Write the number in scientific notation. ^ Solution: Decimal point moves 7 places to the left

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Converting a Positive Number from Scientific Notation Multiplying a positive number by a positive power of 10 makes the number greater, so move the decimal point to the right if n is positive in 10 n. Multiplying a positive number by a negative power of 10 makes the number less, so move the decimal point to the left if n is negative. If n is 0, leave the decimal point where it is. Slide 5.1- 33 Use the rules for exponents with scientific notation.

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Write each number in standard notation. 2.51 ×10 3 –6.8 ×10 −4 = 2510 = –0.000068 Move the decimal 3 places to the right. Move the decimal 4 places to the left. Slide 5.1- 34 CLASSROOM EXAMPLE 10 Converting from Scientific Notation to Standard Notation Solution: When converting from scientific notation to standard notation, use the exponent to determine the number of places and the direction in which to move the decimal point.

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. Evaluate Slide 5.1- 35 CLASSROOM EXAMPLE 11 Using Scientific Notation in Computation Solution:

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Copyright © 2012, 2008, 2004 Pearson Education, Inc. The distance to the sun is 9.3 × 10 7 mi. How long would it take a rocket traveling at 3.2 × 10 3 mph to reach the sun? d = rt, so It would take approximately 2.9 ×10 4 hours. Slide 5.1- 36 CLASSROOM EXAMPLE 12 Using Scientific Notation to Solve Problems Solution:

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