#### Transcript 8.1 Laws of Exponents: Multiplying Monomials

Page 370 – Laws of Exponents Multiplying Monomials 8.1 Laws of Exponents: Multiplying Monomials Objectives Define exponents and powers. Find products of powers. Simplify products of monomials. 8.1 Laws of Exponents: Multiplying Monomials Glossary Terms base of a power coefficient exponent monomial Product-of-Powers Property Base and Exponent exponent 4 base 3 The exponent tells us how many times the base is used as a factor. Rules and Properties Exponents xm = x x x . . . x m factors For all real numbers x and all positive integers m, when m = 1, xm = x1 = x. Evaluate 28 = 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 256 = 3 5 = 5 · 5 · 5 = 125 34 = 3 · 3 · 3 · 3 = 81 61 = 6 8.1 Laws of Exponents: Multiplying Monomials Rules and Properties Product-of-Powers Property For all nonzero real numbers x and all integers m and n, xm xn = xm+n When you multiply numbers with the same base, add the exponents. Simplify 23 · 24 = 2 · 2 · 2 · 2 · 2 · 2 · 2 = 27 = 128 8 · 8³ = 8 · 8 · 8 · 8 = 84 = 4096 y2 · y5 = y2 + 5 = y7 5m · 5p = 5m + p Suppose that a colony of bacteria doubles in size every hour. If the colony contains 1000 bacteria at noon, how many bacteria will the colony contain at 3 p.m. and 5 p.m. of the same day? Between noon and 3 p.m. there are 3 hours so there will be 1000 · 2³, or 8000 bacteria. The 2 stands for the doubling and the 3 stands for 3 hours. At 5 p.m., 2 hours later, the bacteria will double 2 more times. There will be (1000 · 2³) · 2², or 1000 · 23 + 2, or 1000 · 25, 32,000 bacteria in the colony. 8.1 Laws of Exponents: Multiplying Monomials Rules and Properties Definition of Monomial monomial: a constant, variable, or a product of a constant and one or more variables Coefficient – the number that goes with the variable To multiply monomials 1. 2. Remove the parentheses and use the commutative and associative properties to rearrange the terms. Group the constants together, and then group like terms together. Simplify by using the Product-of-Powers Property when appropriate. Simplify (5t)(-30t²) = 5 · (-30) · t · t² = -150t³ (-4a²b)(-ac²)(3b²c²) = -4 · (-1) · 3 · a² ·a · b · b² · c² · c² = 12a³b³c4 (3m²)(60mp²) = 3 · 60 · m² · m · p² = 180m³p² (8xz)(-10y)(-2yz²) = 8 · (-10) · (-2) · x · y · y ·z · z² = 160xy²z³ 8.1 Laws of Exponents: Multiplying Monomials Key Skills Simplify the product of monomials containing exponents. Simplify (5c2d3)(7cd5) Group terms Multiply the with the same base. constants. = 35 c2 c d3 d5 Use the Product-ofPowers Property. = 35 c2 + 1 d3 + 5 Simplify = 35c3d8 TOC The volume of a right rectangular prism can be found by using the formula V = lwh. Suppose a prism has a length of 2xy, a width of 3xy, and a height of 6xyz. Find the volume. V = lwh = (2xy)(3xy)(6xyz) =2·3·6·x·x·x·y·y·y·z = 36x³y³z Assignment Page 374 – 376 – # 10 – 50 even, 52 – 55, 61 – 64, 65 - 73