1. Multiplying Monomials and Raising Monomials to Powers

2. 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.

3. Writing - Using Exponents
Rewrite the following expressions using exponents:
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.

4. Writing Expressions without Exponents
Write out each expression without exponents (as multiplication): or

5. Product of Powers Simplify the following expression: (5a2)(a5)
There are two monomials. Underline them.
What operation is between the two monomials?
Multiplication!
Step 1: Write out the expressions in expanded form.
Step 2: Rewrite using exponents.

6. Product of Powers Rule am • an = am+n.
For any number a, and all integers m and n,
am • an = am+n.

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

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

9. The monomial is the term inside the parentheses.
Power of Powers
Simplify the following: ( x3 ) 4
The monomial is the term inside the parentheses.
Step 1: Write out the expression in expanded form.
Step 2: Simplify, writing as a power.
Note: 3 x 4 = 12.

10. Power of Powers Rule
For any number, a, and all integers m and n,

11. Monomials to Powers
If the monomial inside the parentheses has a coefficient, raise the coefficient to the power, but still multiply the variable powers.

12. Monomials to Powers (Power of a Product)
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

13. Monomials to Powers (Power of a Product)
Simplify each expression:

14. Dividing Monomials

15. Let's look at the rules.
For all integers “m” and “n” and any nonzero number “a” ……
When the problems look like this, and the bases are the same, you will subtract the exponents.
ANY number raised to the zero power is equal to ONE.
If the exponent is negative, it is written on the wrong side of the fraction bar, move it to the other side.

16. 1.
Subtract the exponents 2.
When dividing the coefficients or subtracting the exponents, place the answer on the side of the fraction bar that has the larger value for that term.
Divide the coefficients

17. If you’re having trouble with the rules, try it like this………
First, factor numerator and denominator into prime factors.
Second, cross out like terms from top to bottom. (reduce)
Third, list remaining terms and multiply them back together.

18. U’s cancel
Each other 3.
Subtract the exponents 4.
When dividing the coefficients or subtracting the exponents, place the answer on the side of the fraction bar that has the larger value for that term.
Divide the coefficients

19. 5.
Remember, if the exponent is negative, the term is written on the wrong side of the fraction bar, move it to the other side. 6.
Now
Subtract
The
Exponents

20. Now divide the coefficients but now ADD the exponents
Fix your
negative
exponent 7. 8.
ANY number raised to the zero power is equal to ONE.

21. Now divide the coefficients and combine the exponents
Fix your
negative
exponents 9.

22. And INSIDE …… Distribute!!
Exponents OUTSIDE
And INSIDE …… Distribute!!
10.
11.

23. First – Simplify the numerator!!
12.
Then the
denominator
Now
Subtract
The
Exponents

24. First – Simplify the numerator!!
13.
Then the
denominator
Now
Subtract
The
Exponents

25. And INSIDE …… Distribute!!
Exponents OUTSIDE
And INSIDE …… Distribute!!
14.
Fix your
Negative exponents
Now
Subtract
The
Exponents

26. Steps Multiply exponents Fix negative exponents
Any neg. number raised to even power becomes positive…
Subtract exponents (unless they are beside one another—then add)

27. And INSIDE …… Distribute!!
Exponents OUTSIDE
And INSIDE …… Distribute!!
15.
Fix your
Negative exponents
Now
Subtract
The
Exponents

28. 16.

29. 17.

30. Multiply exponents

31. And INSIDE …… Distribute!!
Exponents OUTSIDE
And INSIDE …… Distribute!!
Any neg.
number
raised
to an
even power
is positive
Use ( ) when
Typing the -4
Into calculator!