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:-9 y 7a 3y3 ½abc5

• These are NOT monomials:

m + n x/y 3 – 4b 1/x2 7y/9z

Notes:

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

20 21 22 23 24 25 26 27 28 29 210

0 2 4 8 16 32 64 128 256 512 1024

Notes:

• Notice that each of the following is true:4 · 16 = 64 8 · 16 = 128 8 · 32 = 256

22 · 24 = 26 23 · 24 = 27 23 · 25 = 28

Look for a pattern in the products shown. 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 a and all integers m and n,

am · an = am+n

Ex. 1: Find the measure of the area of a rectangle.

A = lw= x3 · x4

= x3+4

= x7x3

x4

Ex. 2: Simplify (-5x2)(3x3y2)( xy4)

(-5x2)(3x3y2)( xy4) = (-5· 3 · )(x2 · x3 · x)(y2 · y4)

= -6x2+3+1y2+4

= -6x6y6

Step 1: Commutative and associative propertiesStep 2: Product of Powers PropertyStep 3: Simplify

2

5

2

5

2

5

Notes: Take a look at the examples below:

(52)4 = (52)(52)(52)(52)

= 52+2+2+2

= 58

(x6)2 = (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 a and all integers m and n,

(am)n = amn

Here are a few more examples

(xy)3 = (xy)(xy)(xy) = (x · x · x)(y · y · y) = x3y3

(4ab)4 = (4ab) (4ab) (4ab) (4ab) = (4 · 4 · 4 ·4)(a · a · a · a)(b · b · b · b) = 44a4b4

= 256a4b4

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 = ambm

Ex. 3—Find the measure of the volume of the cube.

V = s3

= (x2y4)3

= (x2)3 · (y4)3

= x 2·3y4·3

= x6y12

x2y4

x2y4

x2y4

Power of a Monomial

• For any number a and b, and any integers m, n, and p,

(ambn)p = ampbnp

Ex. 4: Simplify (9b4y)2[(-b)2]3

(9b4y)2[(-b)2]3 = 92(b4)2y2(b2)3

= 81b8y2b6

= 81b14y2

Some calculators have a power key labeled yx . 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: 0.002744, so (0.14)3 is about 0.003

yx 3 =