CHAPTER 5 Number Theory and the Real Number
CHAPTER 5 Number Theory and the Real Number System Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 5. 6, Slide 1
5. 6 Exponents and Scientific Notation Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 5. 6, Slide 2
1. 2. 3. 4. 5. Objectives Use properties of exponents. Convert from scientific notation to decimal notation. Convert from decimal notation to scientific notation. Perform computations using scientific notation. Solve applied problems using scientific notation. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 5. 6, Slide 3
Properties of Exponents Property Meaning The Product Rule bm · b n = bm + n When multiplying exponential 96 · 912 = 96 + 12 expressions with the same base, = 918 add the exponents. Use this sum as the exponent of the common base. The Power Rule (bm)n = bmn When an exponential expression is (34)5 = 34· 5 = 320 raised to a power, multiply the (53)8 = 53· 8 = 524 exponents. Place the product of the exponents on the base and remove the parentheses. The Quotient Rule When dividing exponential expressions with the same base, subtract the exponent in the denominator from the exponent in the numerator. Use this difference as the exponent of the common base. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Examples Section 5. 6, Slide 4
The Zero Exponent Rule If b is any real number other than 0, b 0 = 1. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 5. 6, Slide 5
Example: Using the Zero Exponent Rule Use the zero exponent rule to simplify: a. 70=1 b. c. ( 5)0 = 1 d. 50 = 1 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 5. 6, Slide 6
The Negative Exponent Rule If b is any real number other than 0 and m is a natural number, Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 5. 6, Slide 7
Example: Using the Negative Exponent Rule Use the negative exponent rule to simplify: a. b. c. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 5. 6, Slide 8
Powers of Ten 1. A positive exponent tells how many zeros follow the 1. For example, 109, is a 1 followed by 9 zeros: 1, 000, 000. 2. A negative exponent tells how many places there are to the right of the decimal point. For example, 10 -9 has nine places to the right of the decimal point. 10 -9 = 0. 00001 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 5. 6, Slide 9
Scientific Notation A positive number is written in scientific notation when it is expressed in the form a 10 n , where a is a number greater than or equal to 1 and less than 10 (1 ≤ a < 10), and n is an integer. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 5. 6, Slide 10
Convert Scientific Notation to Decimal Notation If n is positive, move the decimal point in a to the right n places. If n is negative, move the decimal point in a to the left |n| places. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 5. 6, Slide 11
Example: Converting from Scientific to Decimal Notation Write each number in decimal notation: a. 1. 375 1010 b. 1. 1 10 -4 In each case, we use the exponent on the 10 to move the decimal point. In part (a), the exponent is positive, so we move the decimal point to the right. In part (b), the exponent is negative, so we move the decimal point to the left. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 5. 6, Slide 12
Converting From Decimal to Scientific Notation To write the number in the form a 10 n: Determine a, the numerical factor. Move the decimal point in the given number to obtain a number greater than or equal to 1 and less than 10. Determine n, the exponent on 10 n. The absolute value of n is the number of places the decimal point was moved. The exponent n is positive if the given number is greater than or equal to 10 and negative if the given number is between 0 and 1. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 5. 6, Slide 13
Example: Converting from Decimal Notation to Scientific Notation Write each number in scientific notation: a. 4, 600, 000 b. 0. 000023 Solution: Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 5. 6, Slide 14
Computations with Scientific Notation We use the product rule for exponents to multiply numbers in scientific notation: (a 10 n) (b 10 m) = (a b) 10 n+m Add the exponents on 10 and multiply the other parts of the numbers separately. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 5. 6, Slide 15
Example: Multiplying Numbers in Scientific Notation Multiply: (3. 4 109)(2 10 -5). Write the product in decimal notation. Solution: (3. 4 109)(2 10 -5) = (3. 4 2)(109 10 -5) = 6. 8 109+(-5) = 6. 8 104 = 68, 000 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 5. 6, Slide 16
Computations with Scientific Notation We use the quotient rule for exponents to divide numbers in scientific notation: Subtract the exponents on 10 and divide the other parts of the numbers separately. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 5. 6, Slide 17
Example: Dividing Numbers In Scientific Notation Divide: . Write the quotient in decimal notation. Solution: Regroup factors. Subtract the exponents. Write the quotient in decimal notation. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 5. 6, Slide 18
Example: The National Debt As of December 2011, the national debt was $15. 2 trillion, or 15. 2 1012 dollars. At that time, the U. S. population was approximately 312, 000, or 3. 12 108. If the national debt was evenly divided among every individual in the United States, how much would each citizen have to pay? Solution: The amount each citizen would have to pay is the total debt, 15. 2 1012, divided among the number of citizens, 3. 12 108. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 5. 6, Slide 19
Example: The National Debt continued Every citizen would have to pay approximately $48, 700 to the federal government to pay off the national debt. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 5. 6, Slide 20
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