Dispatch a · a · a · a 2 · 2 · 2 x · x · x · y · y · z · z · z 32x3 Rewrite the following expressions in exponential form: a · a · a · a 2 · 2 · 2 x · x · x · y · y · z · z · z Rewrite in expanded form: 32x3 52xy2z3

Quick Review Rewrite the following expressions using exponent form: The variables, x and y, represent the bases. The number of times each base is multiplied by itself will be the value of the exponent.

Multiplying Monomials

Standard/Objectives Standard: 2.0 Objectives: After studying this lesson, you should be able to: Multiply monomials, and Simplify expressions involving powers of monomials.

Vocabulary Monomials - a number, a variable, or a product of a number and one or more variables 4x, 20x2yw3, -3, a2b3, and 3yz are all monomials. Constant (coefficient) – a monomial that is a number without a variable. Base – In an expression of the form xn, the base is x. Exponent – In an expression of the form xn, the exponent is n.

Are these monomials? -9 3y3 3 – 4b x/y ½abc5 7+a Yes No

Look for a pattern in the products shown a2 · a4 = a6 a3 · a4 = a7 a3 · a5 = a8 If you consider only the exponents, you will find that 2 + 4 = 6, 3 + 4 = 7, and 3 + 5 = 8 These examples suggest that you can multiply powers that have the same base by adding exponents.

Product of Powers Property For any number x and all integers m and n, xm · xn = xm+n

Simplify: (2m2)(3m3) Step 1: Multiply coefficients Answer: 6m 5 Step 1: Multiply coefficients Step 2 : Add exponents

Your Turn Simplify: (y 2 z)(yz 2 ) Step 1: Multiply coefficients Step 2 : Add exponents

Look for a pattern to these problems. (52)4 = 58 = (52)(52)(52)(52) = 52+2+2+2 = 58 (x6)2 = x12 = (x6)(x6) = x6+6 = x12 Since (52)4 = 58 and (x6)2 = x12, these examples suggest that you can find the power of a power by multiplying exponents.

Power of a power For any number x and all integers m and n, (xm)n = xmn

Simplify the following: (-22) 5

Your Turn Simplify: (-52) 6

Try this What do you notice about these problems? (xy)3 (-4x3y2)2

= (4 · 4 · 4 ·4)(a · a · a · a)(b · b · b · b) = 44a4b4 = 256a4b4 Simplify the following: (4ab)4 = (4ab) (4ab) (4ab) (4ab) = (4 · 4 · 4 ·4)(a · a · a · a)(b · b · b · b) = 44a4b4 = 256a4b4 Distribute the outside exponent with the inside exponents ONLY. (Multiply each exponent) Simplify.

Power of a Product For any number x and all integers m, (xy)m = xmym

Your Turn Simplify: (3ab)3 Distribute the outside exponent to the inside exponents ONLY. (Multiply each exponent) Simplify.

Simplify: (3p3q2) 2 Step 1: Distribute outside exponent to the inside exponents. Step 2 : Simplify.

Your Turn Simplify: (2b3c4) 2