 # CHAPTER 4 Polynomials: Operations Slide 2Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. 4.1Integers as Exponents 4.2Exponents and Scientific.

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CHAPTER 4 Polynomials: Operations Slide 2Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. 4.1Integers as Exponents 4.2Exponents and Scientific Notation 4.3Introduction to Polynomials 4.4Addition and Subtraction of Polynomials 4.5Multiplication of Polynomials 4.6Special Products 4.7Operations with Polynomials in Several Variables 4.8Division of Polynomials

OBJECTIVES 4.2 Exponents and Scientific Notation Slide 3Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. aUse the power rule to raise powers to powers. bRaise a product to a power and a quotient to a power. cConvert between scientific notation and decimal notation. dMultiply and divide using scientific notation. eSolve applied problems using scientific notation.

For any real number a and any integers m and n, To raise a power to a power, multiply the exponents. 4.2 Exponents and Scientific Notation The Power Rule Slide 4Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE a)(x 3 ) 4 b) (4 2 ) 8 c) (a -3 ) -2 Solution a) (x 3 ) 4 = x 3  4 = x 12 b) (4 2 ) 8 = 4 2  8 = 4 16 c) (a –3 ) –2 = a (–3)(–2) = a 6 4.2 Exponents and Scientific Notation a Use the power rule to raise powers to powers. ASimplify: Slide 5Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

For any real number a and b and any integer n, To raise a product to the nth power, raise each factor to the nth power. 4.2 Exponents and Scientific Notation Raising a Product to a Power Slide 6Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE a) (3x) 4 b) (  2x 3 ) 2 c) (a 2 b 3 ) 7 (a 4 b 5 ) d) (3x 4 y -5 ) 4 e) (  2x -3 y 5 ) -3 Solution a) (3x) 4 = 3 4 x 4 = (3 3 3 3)x 4 = 81x 4 b) (  2x 3 ) 2 = (  2) 2 (x 3 ) 2 Raising each factor to the 2 nd power = 4x 6 4.2 Exponents and Scientific Notation b Raise a product to a power and a quotient to a power. BSimplify: (continued) Slide 7Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE a) (3x) 4 b) (  2x 3 ) 2 c) (a 2 b 3 ) 7 (a 4 b 5 ) Solution c) (a 2 b 3 ) 7 (a 4 b 5 ) = (a 2 ) 7 (b 3 ) 7 a 4 b 5 = a 14 b 21 a 4 b 5 Multiplying exponents = a 18 b 26 Adding exponents 4.2 Exponents and Scientific Notation b Raise a product to a power and a quotient to a power. BSimplify: (continued) Slide 8Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE d) (3x 4 y –5 ) 4 e) (  2x –3 y 5 ) –3 Solution d) (3x 2 y –5 ) 4 = (3 4 )(x 4 ) 4 (y –5 ) 4 4.2 Exponents and Scientific Notation b Raise a product to a power and a quotient to a power. BSimplify: (continued) Slide 9Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE d) (3x 4 y –5 ) 4 e) (  2x –3 y 5 ) –3 Solution e) (  2x –3 y 5 ) –3 = (–2 ) –3 (x –3 ) –3 (y 5 ) –3 4.2 Exponents and Scientific Notation b Raise a product to a power and a quotient to a power. BSimplify: Slide 10Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

For any real numbers a and b, b ≠ 0, and any integer n, To raise a quotient to the nth power, raise numerator AND denominator to the nth power. Also, 4.2 Exponents and Scientific Notation Raising a Quotient to a Power Slide 11Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE a) b) c) 4.2 Exponents and Scientific Notation b Raise a product to a power and a quotient to a power. CSimplify: Slide 12Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

Scientific notation for a number is an expression of the type where n is an integer, M is greater than or equal to 1 and less than 10 (1 ≤ M < 10), and M is expressed in decimal notation. 10 n is also considered to be scientific notation when M = 1. 4.2 Exponents and Scientific Notation Scientific Notation Slide 13Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

Scientific notation A positive exponent indicates a large number (greater than or equal to 10). A negative exponent indicates a small number (between 0 and 1). 4.2 Exponents and Scientific Notation c Convert between scientific notation and decimal notation. Slide 14Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE a) 94,000b) 0.0423 Solution a) 94,000 = 9.4  10 4 b) 0.0423 0.04.23 4 places to the left A Large number so the exponent is positive. 2 places to the right Small number so the exponent is negative. 4.2 Exponents and Scientific Notation c Convert between scientific notation and decimal notation. AConvert to scientific notation: Slide 15Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. We want a number between 1 and 10 so we must move the decimal point 94,000.. = 4.23  10  2

EXAMPLE a) 3.842  10 6 b) 5.3  10  7 Solution a) 3.842  10 6 = 3.842000  10 6 = 3,842,000. b) 5.3  10  7 = 0.00000053 = 0.0000005.3 6 places to the right Positive exponent, so the answer is a large number. 7 places Negative exponent, so the answer is a small number. 4.2 Exponents and Scientific Notation c Convert between scientific notation and decimal notation. BConvert to decimal notation: Slide 16Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE Solution (1.7 × 10 8 )(2.2 × 10 –5 ) = (1.7 · 2.2) · (10 8 · 10 –5 ) = 3.74 × 10 8 +(–5) = 3.74 × 10 3 4.2 Exponents and Scientific Notation d Multiply and divide using scientific notation. CSimplify: (1.7 × 10 8 )(2.2 × 10 –5 ) Slide 17Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE Solution (6.2  10  9 )  (8.0  10 8 ) = 4.2 Exponents and Scientific Notation d Multiply and divide using scientific notation. D Simplify. (6.2  10  9 )  (8.0  10 8 ) Slide 18Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE Solution: We multiply one billion by 3500. (1.0  10 9 )  (3.5  10 3 ) = 3.5  10 12 4.2 Exponents and Scientific Notation e Solve applied problems using scientific notation. EA gigabyte is a measure of a computer’s storage capacity. One gigabyte holds about one billion bytes of information. If a firm’s computer network contains 3500 gigabytes of memory, how many bytes are in the network? Slide 19Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

4.2 Exponents and Scientific Notation Definitions and Rules for Exponents (continued) Slide 20Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

4.2 Exponents and Scientific Notation Definitions and Rules for Exponents Slide 21Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

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