Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Chapter 4 Polynomials

1-2 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Exponents and Their Properties Multiplying Powers with Like Bases Dividing Powers with Like Bases Zero as an Exponent Raising a Power to a Power Raising a Product or a Quotient to a Power 4.1

Copyright © 2014, 2010, and 2006 Pearson Education, Inc. The Product Rule For any number a and any positive integers m and n, (To multiply powers with the same base, keep the base and add the exponents.)

1-4 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Multiply and simplify each of the following. (Here “simplify” means express the product as one base to a power whenever possible.) a) x 3  x 5 b) 6 2  6 7  6 3 c) (x + y) 6 (x + y) 9 d) (w 3 z 4 )(w 3 z 7 ) Solution a) x 3  x 5 = x 3+5 Adding exponents = x 8

1-5 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example b) 6 2  6 7  6 3 c) (x + y) 6 (x + y) 9 d) (w 3 z 4 )(w 3 z 7 ) Solution b) 6 2  6 7  6 3 = = 6 12 c) (x + y) 6 (x + y) 9 = (x + y) 6+9 = (x + y) 15 d) (w 3 z 4 )(w 3 z 7 ) = w 3 z 4 w 3 z 7 = w 3 w 3 z 4 z 7 = w 6 z 11

Copyright © 2014, 2010, and 2006 Pearson Education, Inc. The Quotient Rule For any nonzero number a and any positive integers m and n for which m > n, (To divide powers with the same base, subtract the exponent of the denominator from the exponent of the numerator.)

1-7 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Divide and simplify each of the following. (Here “simplify” means express the product as one base to a power whenever possible.) a)b)c)d) Solution a) b)

1-8 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example c) d)

Copyright © 2014, 2010, and 2006 Pearson Education, Inc. The Exponent Zero For any real number a, with a ≠ 0, (Any nonzero number raised to the 0 power is 1.)

1-10 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Simplify: a) b) (  3) 0 c) (4w) 0 d) (  1)8 0 e)  9 0. Solution a) = 1 b) (  3) 0 = 1 c) (4w) 0 = 1, for any w  0. d) (  1)8 0 = (  1)1 =  1 e)  9 0 is read “the opposite of 9 0 ” and is equivalent to (  1)9 0 :  9 0 = (  1)9 0 = (  1)1 =  1

Copyright © 2014, 2010, and 2006 Pearson Education, Inc. The Power Rule For any number a and any whole numbers m and n, (a m ) n = a mn. (To raise a power to a power, multiply the exponents and leave the base unchanged.)

1-12 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Simplify: a)(x 3 ) 4 b) (4 2 ) 8 Solution a) (x 3 ) 4 = x 3  4 = x 12 b) (4 2 ) 8 = 4 2  8 = 4 16

Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Raising a Product to a Power For any numbers a and b and any whole number n, (ab) n = a n b n. (To raise a product to a power, raise each factor to that power.)

1-14 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Simplify: a)(3x) 4 b) (  2x 3 ) 2 c) (a 2 b 3 ) 7 (a 4 b 5 ) Solution a) (3x) 4 = 3 4 x 4 = 81x 4 b) (  2x 3 ) 2 = (  2) 2 (x 3 ) 2 = 4x 6 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

Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Raising a Quotient to a Power For any real numbers a and b, b ≠ 0, and any whole number n, (To raise a quotient to a power, raise the numerator to the power and divide by the denominator to the power.)

1-16 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Simplify: a)b) c) Solution a) b) c)

Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Definitions and Properties of Exponents For any whole numbers m and n, 1 as an exponent:a 1 = a 0 as an exponent:a 0 = 1 The Product Rule: The Quotient Rule: The Power Rule:(a m ) n = a mn Raising a product to a power:(ab) n = a n b n Raising a quotient to a power: