Multiplying Monomials

Objectives Be able to multiply monomials. Be able to simplify expressions involving powers of monomials.

Vocabulary Monomials - a number, a variable, or a product of a number and one or more variables. Constant – A monomial that is a real number. Power – An expression in the form xn. Base – In an expression of the form xn, the base is x. Exponent – In an expression of the form xn, the exponent is n. Exponent Power Base

Writing Using Exponents Rewrite the following expressions using exponents. The variables, x and y, represent the bases. The number of times each base occurs will be the value of the exponent. è 3 4 ø 2 æ ç ö ÷ 2 3 2 3 2 3 2 3 · · · =

Writing Out Expressions with Exponents Write out each expression the long way. The exponent tells how many times the base occurs. If the exponent is outside the parentheses, then the exponent belongs with each number and/or variable inside the parentheses.

) ( 5 · a = 5 a 5 a = 5 · a · a · a · a · a · a · a Product of Powers Simplify the following expression: (5a2)(a5). Step 1: Write out the expressions the long way or in expanded form. ) ( 5 2 a = 5 · a · a · a · a · a · a · a Step 2: Rewrite using exponents. 5 · a = 7 5 a For any number a, and all integers m and n, am • an = am+n

) ( Power of a Power x = x · x · x · x Simplify the following: First, write the expression in expanded form. ) ( x 3 4 = x 3 · x 3 · x 3 · x 3 However, Therefore, Note: 3 x 4 = 12. For any number, a, and all integers m and n,

( ) Power of a Product xy = xy · xy · xy · xy · xy = x · ( )( · y ) = Simplify: (xy)5 ( ) xy 5 = xy · xy · xy · xy · xy = x · ( )( · y ) = x y 5 For all numbers a and b, and any integer m,

( ( ) ) Example Problems · 4 4 4 ( ) a a a 3 x y 3 x y 27 x y 5 6 5 6 Simplify: ) ( 3 4 x y ( ) a 3 6 4 5 6 · ) ( 3 4 x y a 3 6 · 4 5 6 + a 18 4 11 27 12 3 x y Apply the Power of a Product Property and Simplify. Apply the Product of Powers property Apply the Power of a Power Property.

You Try It ( ) 1 4 5 7 3 . r t ( ) 2 1 3 4 . w æ è ç ö ø ÷

) ( ) ( ( ) Problem 1 r t r t · = r t = = r t 4 5 7 3 4 7 5 3 4 7 5 3 + = = r t 11 8 Group like terms. Apply the Product of Powers Property.

Problem 2