Chapter 4 Polynomials
Exponents and Their Properties 4.1 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
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.)
Multiply and simplify each of the following Multiply and simplify each of the following. (Here “simplify” means express the product as one base to a power whenever possible.) a) x3 x5 b) 62 67 63 c) (x + y)6(x + y)9 d) (w3z4)(w3z7) Solution a) x3 x5 = x3+5 Adding exponents = x8
(cont) b) 62 67 63 c) (x + y)6(x + y)9 d) (w3z4)(w3z7) Solution = 612 c) (x + y)6(x + y)9 = (x + y)6+9 = (x + y)15 d) (w3z4)(w3z7) = w3z4w3z7 = w3w3z4z7 = w6z11
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.)
Divide and simplify each of the following Divide and simplify each of the following. (Here “simplify” means express the quotient as one base to a power whenever possible.) a) b) c) d) Solution a) b)
(cont) c) d)
The Exponent Zero For any real number a, with a ≠ 0, (Any nonzero number raised to the 0 power is 1.)
Simplify: a) 12450 b) (3)0 c) (4w)0 d) (1)80 e) 90. Solution a) 12450 = 1 b) (3)0 = 1 c) (4w)0 = 1, for any w 0. d) (1)80 = (1)1 = 1 e) 90 is read “the opposite of 90” and is equivalent to (1)90: 90 = (1)90 = (1)1 = 1
The Power Rule For any number a and any whole numbers m and n, (am)n = amn. (To raise a power to a power, multiply the exponents and leave the base unchanged.)
Simplify: a) (x3)4 b) (42)8 Solution a) (x3)4 = x34 = x12 b) (42)8 = 428 = 416
Raising a Product to a Power For any numbers a and b and any whole number n, (ab)n = anbn. (To raise a product to a power, raise each factor to that power.)
Simplify: a) (3x)4 b) (2x3)2 c) (a2b3)7(a4b5) Solution a) (3x)4 = 34x4 = 81x4 b) (2x3)2 = (2)2(x3)2 = 4x6 c) (a2b3)7(a4b5) = (a2)7(b3)7a4b5 = a14b21a4b5 Multiplying exponents = a18b26 Adding exponents
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.)
Simplify: a) b) c) Solution a) b) c)
Definitions and Properties of Exponents For any whole numbers m and n, 1 as an exponent: a1 = a 0 as an exponent: a0 = 1 The Product Rule: The Quotient Rule: The Power Rule: (am)n = amn Raising a product to a power: (ab)n = anbn Raising a quotient to a power: