Monomials

Multiplying & Dividing Monomials

and Raising Monomials to Powers

Today:

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 – 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.

Writing Expressions Using Exponents

Write the expression with exponents

(as multiplication):

8a3b38 ● a ● a ● a ● b ● b ● b =Could the above expression be

written as a power of a product? ( )x

x x x x y y y y

xy xy xy xy xy 4or

Simplify the following expression: (5a2)(a5)

Step 1: Write out the expressions in expanded form.

Step 2: Rewrite using exponents.

Product Rule

5a2 a5 5 a a a a a a a

How many terms are there?

What operation is being performed? Multiplication!

5a2 a5 5 a

7 5a

7

Multiplying Monomials: The Product Rule

4) 3k5mn

4 7k3m

3n

3

5) 12 x2y

3 2xy2 24x

3y

5

21k8m

4n

7

If the monomials have coefficients, multiply those, but still add powers of common bases.

If the monomial inside the parentheses has more than one variable, raise each variable to the outside power using the power of a power rule.

(ab)m = am•bm

(9xy)2 = (-5x)2 = -(5x)2 =

Simplify the following: ( x3 ) 4

Note: 3 x 4 = 12

The monomial is the term inside the parentheses.

1. Multiply the exponents, write the simplified monomial

x3

4

x12

For any number, a, and all integers m and n,

am n

amn .

1) b9

10

b90

2) c3

3

c9

1) 2b9

3

8b27

2) 5c3

3

125c9

3) 7w12

2

49w24

If the monomial inside the parentheses has a coefficient, raise the coefficient to the power, but still

multiply the variable powers.

Dividing Monomials

For all integers “m” and “n” and any nonzero number “a” ……

Let's review the rules.

m

n

a

a

m na When the problems look like this, and the bases are the same, you will subtract the exponents.

0 1a ANY number raised to the zero power is equal to ONE.

na 1na

If the exponent is negative, it is written on the wrong side of the fraction bar, move it to the other side, and change the sign.

1. 3 2 2f g h

fgh

3 1 2 1 2 1f g h 2 1 1f g h

2. 3 5

7

24

6

x y

xy

Subtract the exponents

42x

2y

Reminder: Never finish a problem with negative

exponents

3. 0 4 2

2 3 2

5 t wu

t w u

1

4. 4 5

2 6

27

9

x y

x y

Subtract the exponents

3 2xy

U’s cancelEach other

2t2w