1. LAWS OF EXPONENTS

2. 22 = 2 X 2 X 2 X 2 Basic Terminology
Exponent – In exponential notation, the number of times the base is used as a factor.
Base – In exponential notation, the number or variable that undergoes repeated MULTIPLICATION.
EXPONENT
22 = 2 X 2 X 2 X 2
BASE
Review the vocabulary. The base and the exponent together is called the power.

3. IMPORTANT EXAMPLES -34 means –(3 x 3 x 3 x 3) = -81
Very important to stress that the negative sign only goes with the number if it is included in the parentheses.
(-3)3 means (–3) x (–3) x (–3) = -27

4. Try these:
– 24
– 52
(– 5)2

5. x5 = (x)(x)(x)(x)(x) y3 = (y)(y)(y) Variable Expressions
Explain that using parentheses is the same as multiplication.

6. Substitution and Evaluating
STEPS
Write out the original problem.
Show the substitution with parentheses.
Work out the problem.
Solve x3 if x = 4 x3
43 = 64

7. Evaluate the variable expression when x = 1, y = 2, and w = -3
Step 1
Step 2
Step 3

8. Evaluate the variable expression when x = 1, y = 2, and w = -3
Step 1
Step 2
Step 3

9. Evaluate the variable expression when x = 1, y = 2, and w = -3
Step 1
Step 2
Walk the partiicpants through each step.
Step 3

10. MULTIPLICATION PROPERTIES
PRODUCT OF POWERS
This property is used to combine 2 or more exponential expressions with the SAME base.
a3 x a4 = (a x a x a) x (a x a x a x a)
= a7
d3d6 = (d x d x d) x (d x d x d x d x d x d)
= a3+6
= a9
23 x 25 = (2 x 2 x 2) x (2 x 2 x 2 x 2 x 2)
= 28
= 256
Ask the participants what they notice about the exponents in the problems and the exponent of the answer. Let them discover that when you multiply powers with the same base you will add the exponents.

11. MULTIPLICATION PROPERTIES
POWER TO A POWER
This property is used to write and exponential expression as a single power of the base.
(52)3 =
(52)(52)(52) = = 56
=b8
(b2)4 =
(b2)(b2)(b2)(b2) =
b =
b(2x4)
Explain that 5 squared is raised to the third power. Which means 5 squared is multiplied times itself 3 times.
Do several more examples ((34)5 = 320, (y2)7 = y14
Help the participants discover that when you raise a power to a power you multiply the exponents.
(43)5 =
43x5 =
415

12. MULTIPLICATION PROPERTIES
POWER OF PRODUCT
This property combines the first 2 multiplication properties to simplify exponential expressions.
(-6 x 5)2 =
(-6)2 x (5)2=
Explain that 5 squared is raised to the third power. Which means 5 squared is multiplied times itself 3 times.
Do several more examples ((34)5 = 320, (y2)7 = y14
Help the participants discover that when you raise a power to a power you multiply the exponents.
36 x 25 = 900
900

13. MULTIPLICATION PROPERTIES
POWER OF PRODUCT
This property combines the first 2 multiplication properties to simplify exponential expressions.
(5xy)3 =
Explain that 5 squared is raised to the third power. Which means 5 squared is multiplied times itself 3 times.
Do several more examples ((34)5 = 320, (y2)7 = y14
Help the participants discover that when you raise a power to a power you multiply the exponents.
(53)(x3)(y3)=
125x3y3

14. MULTIPLICATION PROPERTIES
POWER OF PRODUCT
(4a2)3 x a5=
(43)(a2)3 x a5=
(64)(a2x3) x a5=
Explain that 5 squared is raised to the third power. Which means 5 squared is multiplied times itself 3 times.
Do several more examples ((34)5 = 320, (y2)7 = y14
Help the participants discover that when you raise a power to a power you multiply the exponents.
(64)(a6) x a5=
(64)(a6+5)
64a11

15. 1. (2xy)3 x2 2. (2x3y)4 x4y5 3. (3x2y3)2 (4x2)2 4. (2x)2 (3x+y)2
Simplify each expression.
1. (2xy)3 x2
2. (2x3y)4 x4y5
3. (3x2y3)2 (4x2)2
4. (2x)2 (3x+y)2

16. MULTIPLICATION PROPERTIES PRODUCT OF POWERS add the exponents
SUMMARY
ya x yb = ya+b
PRODUCT OF POWERS add the exponents
POWER TO A POWER
multiply the exponents
(ma)b = ma•b
Review
POWER OF PRODUCT
distribute the exponents
(cd)a = cada

17. ZERO AND NEGATIVE EXPONENTS
ANYTHING TO THE ZERO POWER IS 1.
24 = 2 x 2 x 2 x 2 = 16
16 ÷ 2 = 8
8 ÷ 2 = 4
4 ÷ 2 =2
2 ÷ 2 = 1
Dividing by the base number
23 = 2 x 2 x 2 = 8
22 = 2 x 2 = 4
Have the participants do this themselves with other base numbers. Then ask them what would happen if the kept dividing the answers by the base number. Example of this on the next slide.
21 = 2
20 = 1

18. ZERO AND NEGATIVE EXPONENTS
ANYTHING TO THE ZERO POWER IS 1.
16 ÷ 2 = 8
8 ÷ 2 = 4
4 ÷ 2 =2
2 ÷ 2 = 1
Dividing by the base number
33 = 27
32 = 9
31 = 3
30 = 1
3-1=
Have the participants try this with another problem. 54
3-2 =
3-3 =

19. ZERO AND NEGATIVE EXPONENTS
Explain in this example that the -2 is the exponent for the x but in the next example the -2 exponent applies to the 2 and the x because of the parenthesis.

20. ZERO AND NEGATIVE EXPONENTS
Explain in the previous example that the -2 is the exponent for the x but in this example the -2 exponent applies to the 2 and the x because of the parenthesis.

21. Simplify each expression.

22. DIVISION PROPERTIES QUOTIENT OF POWERS
This property is used when dividing two or more exponential expressions with the same base.
Do examples until the participants discover that when dividing powers with the same base you will subtract the exponents.

23. DIVISION PROPERTIES QUOTIENT OF POWERS
This property is used when dividing two or more exponential expressions with the same base.
Do examples until the participants discover that when dividing powers with the same base you will subtract the exponents.

24. DIVISION PROPERTIES QUOTIENT OF POWERS
This property is used when dividing two or more exponential expressions with the same base.
Do examples until the participants discover that when dividing powers with the same base you will subtract the exponents.

25. DIVISION PROPERTIES QUOTIENT OF POWERS
The exponent 4 is distributed to both the numerator and denominator.

26. ZERO, NEGATIVE, AND DIVISION PROPERTIES
REVIEW
ZERO, NEGATIVE, AND DIVISION PROPERTIES
Zero power
Quotient of powers
Negative Exponents
Power of a quotient
Review the properties