MonomialsMonomialsMultiplying Monomials and Raising

Monomials to Powers

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 - Using ExponentsWriting - Using Exponents

Rewrite the following expressions using exponents:

x x x x y y 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.

x x x x y y x y 4 2

Writing Expressions without Exponents

Write out each expression without exponents (as multiplication):

8a3b2 8a a a b b

xy 4 xyxyxyxy

or

xxxxy yyy

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

Step 1: Write out the expressions in expanded form.

Step 2: Rewrite using exponents.

Product of PowersProduct of Powers

5a2 a5 5a a aa a a a

There are two monomials. Underline them.

What operation is between the two monomials?

Multiplication!

5a2 a5 5a7 5a7

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

am • an = am+n.

Product of Powers RuleProduct of Powers Rule

1) a9 a4 a13

2) w2 w10 w12

4) k5 k3 5) x 2 y2

3) r r5 r 6

x2 y2

k8

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

Multiplying MonomialsMultiplying Monomials

1) 4a9 2a4 8a13

2) 7w2 10w10 70w12

4) 3k 5 7k 3 5) 12x2 2y2

3) 2r 3r5 6r 6

24x2 y2

21k8

These monomials have a mixture of different variables. Only add powers of like variables.

Multiplying MonomialsMultiplying Monomials

1) 4a9b3 2a4b 8a13b4

2) 7w2 y5 10w10y2 70w12y7

4) 3k 5mn4 7k3m3n3 5) 12x2 y3 2xy2

3) 2rt3 3r5 6r 6t 3

24x3 y5

21k8m4n7

Simplify the following: ( x3 ) 4

Note: 3 x 4 = 12.

Power of PowersPower of Powers

The monomial is the term inside the parentheses.

Step 1: Write out the expression in expanded form.

x3 4x3 x3 x3 x3

xxx x x x x x xxxx

x3 4x12

Step 2: Simplify, writing as a power.

Power of Powers RulePower of Powers Rule

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

am namn.

1) b9 10 b90

2) c3 3 c9

3) w12 2 w 24

Monomials to PowersMonomials to Powers

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.

Monomials to Powers(Power of a Product)

Monomials to Powers(Power of a Product)

5w3xy2 4 5 4

w3 4x 4

y2 4

625w34 x4 y24

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

625w12x4 y8

Monomials to Powers(Power of a Product)

Monomials to Powers(Power of a Product)

1) 2b9c4 3 8b27c12

2) 5a5c3 3125a15c9

3) 7w12y4z 2 49w24y8z2

Simplify each expression: