1. Dispatch a · a · a · a 2 · 2 · 2 x · x · x · y · y · z · z · z 32x3
Rewrite the following expressions in exponential form:
a · a · a · a
2 · 2 · 2
x · x · x · y · y · z · z · z
Rewrite in expanded form:
32x3
52xy2z3

2. Quick Review Rewrite the following expressions using exponent form:
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.

3. Multiplying Monomials

4. Standard/Objectives Standard: 2.0
Objectives: After studying this lesson, you should be able to:
Multiply monomials, and
Simplify expressions involving powers of monomials.

5. 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 (coefficient) – 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.

6. Are these monomials? -9
3y3
3 – 4b
x/y
½abc5
7+a
Yes No

7. Look for a pattern in the products shown
a2 · a4 = a6 a3 · a4 = a7 a3 · a5 = a8
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.

8. Product of Powers Property
For any number x and all integers m and n,
xm · xn = xm+n

9. Simplify: (2m2)(3m3) Step 1: Multiply coefficients
Answer: 6m 5
Step 1: Multiply coefficients
Step 2 : Add exponents

10. Your Turn Simplify: (y 2 z)(yz 2 ) Step 1: Multiply coefficients
Step 2 : Add exponents

11. Look for a pattern to these problems.
(52)4 = 58
= (52)(52)(52)(52) =
= 58
(x6)2 = x12
= (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.

12. Power of a power For any number x and all integers m and n,
(xm)n = xmn

13. Simplify the following:
(-22) 5

14. Your Turn
Simplify: (-52) 6

15. Try this
What do you notice about these problems?
(xy)3
(-4x3y2)2

16. = (4 · 4 · 4 ·4)(a · a · a · a)(b · b · b · b) = 44a4b4 = 256a4b4
Simplify the following:
(4ab)4
= (4ab) (4ab) (4ab) (4ab)
= (4 · 4 · 4 ·4)(a · a · a · a)(b · b · b · b)
= 44a4b4
= 256a4b4
Distribute the outside exponent with the inside exponents ONLY.
(Multiply each exponent)
Simplify.

17. Power of a Product
For any number x and all integers m,
(xy)m = xmym

18. Your Turn Simplify: (3ab)3
Distribute the outside exponent to the inside exponents ONLY.
(Multiply each exponent)
Simplify.

19. Simplify: (3p3q2) 2
Step 1: Distribute outside exponent to the inside exponents.
Step 2 : Simplify.

20. Your Turn
Simplify: (2b3c4) 2