6.2 Multiplying Monomials CORD Math Mrs. Spitz Fall 2006

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

What’s a monomial? A monomial is a number, a variable, or a product of a number and one or more variables. Monomials that are real numbers are constants. These are monomials: -9y7a3y 3 ½abc 5 These are NOT monomials: m + nx/y3 – 4b1/x 2 7y/9z

Notes: Recall that an expression of the form x n is a power. The base is x and the exponent is n. A table of powers of 2 is shown below:

Notes: Notice that each of the following is true: 4 · 16 = 648 · 16 = 1288 · 32 = · 2 4 = · 2 4 = · 2 5 = 2 8 Look for a pattern in the products shown. If you consider only the exponents, you will find that = 6, = 7, and = 8 These examples suggest that you can multiply powers that have the same base by adding exponents.

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

Ex. 1: Find the measure of the area of a rectangle. A = l w = x 3 · x 4 = x 3+4 = x 7 x3x3 x4x4

Ex. 2: Simplify (-5x 2 )(3x 3 y 2 )( xy 4 ) (-5x 2 )(3x 3 y 2 )( xy 4 ) = (-5· 3 · )(x 2 · x 3 · x)(y 2 · y 4 ) = -6x y 2+4 = -6x 6 y 6 Step 1: Commutative and associative properties Step 2: Product of Powers Property Step 3: Simplify

Notes: Take a look at the examples below: (5 2 ) 4 = (5 2 )(5 2 )(5 2 )(5 2 ) = = 5 8 (x 6 ) 2 = (x 6 )(x 6 ) = x 6+6 = x 12 Since (5 2 ) 4 = 5 8 and (x 6 ) 2 = x 12, these examples suggest that you can find the power of a power by multiplying exponents.

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

Here are a few more examples (xy) 3 = (xy)(xy)(xy) = (x · x · x)(y · y · y) = x 3 y 3 (4ab) 4 = (4ab) (4ab) (4ab) (4ab) = (4 · 4 · 4 ·4)(a · a · a · a)(b · b · b · b) = 4 4 a 4 b 4 = 256a 4 b 4 These examples suggest that the power of a product is the product of the powers.

Power of a Product For any number a and all integers m, (ab) m = a m b m

Ex. 3—Find the measure of the volume of the cube. V = s 3 = (x 2 y 4 ) 3 = (x 2 ) 3 · (y 4 ) 3 = x 2·3 y 4·3 = x 6 y 12 x2y4x2y4 x2y4x2y4 x2y4x2y4

Power of a Monomial For any number a and b, and any integers m, n, and p, (a m b n ) p = a mp b np

Ex. 4: Simplify (9b 4 y) 2 [(-b) 2 ] 3 (9b 4 y) 2 [(-b) 2 ] 3 = 9 2 (b 4 ) 2 y 2 (b 2 ) 3 = 81b 8 y 2 b 6 = 81b 14 y 2 Some calculators have a power key labeled y x. You can use it to find the powers of numbers more easily. See the next slide.

Ex. 5: Evaluate (0.14) 3 Enter: 0.14 Display will read: , so (0.14) 3 is about yxyx 3 =