1. Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu
Chabot Mathematics
§1.6 Exponent Properties
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer

2. 1.5 Review § Any QUESTIONS About Any QUESTIONS About HomeWork
MTH 55
Review §
Any QUESTIONS About
§1.5 → (Word) Problem Solving
Any QUESTIONS About HomeWork
§1.5 → HW-01

3. Exponent PRODUCT Rule
For any number a and any positive integers m and n,
Exponent
Base
In other Words: To MULTIPLY powers with the same base, keep the base and ADD the exponents

4. Quick Test of Product Rule

5. Example  Product Rule
Multiply and simplify each of the following. (Here “simplify” means express the product as one base to a power whenever possible.) a) x3  x5 b) 62  67  63
c) (x + y)6(x + y)9 d) (w3z4)(w3z7)

6. Example  Product Rule Solution a) x3  x5 = x3+5 Adding exponents
Solution b) 62  67  63 =
= 612
Solution c) (x + y)6(x + y)9 = (x + y)6+9
= (x + y)15
Solution d) (w3z4)(w3z7) = w3z4w3z7
= w3w3z4z7
= w6z11
Base is x
Base is 6
Base is (x + y)
TWO Bases: w & z

7. Exponent QUOTIENT Rule
For any nonzero number a and any positive integers m & n for which m > n,
In other Words: To DIVIDE powers with the same base, SUBTRACT the exponent of the denominator from the exponent of the numerator

8. Quick Test of Quotient Rule

9. Example  Quotient Rule
Divide and simplify each of the following. (Here “simplify” means express the product as one base to a power whenever possible.)
a) b)
c) d)

10. Example  Quotient Rule
Solution a)
Base is x
Solution b)
Base is 8
Solution c)
Base is (6y)
Solution d)
TWO Bases: r & t

11. The Exponent Zero For any number a where a ≠ 0
In other Words: Any nonzero number raised to the 0 power is 1
Remember the base can be ANY Number
, 19.19, −86, , anything

12. Example  The Exponent Zero
Simplify: a) b) (−3)0 c) (4w)0 d) (−1)80 e) −80
Solutions
12450 = 1
(−3)0 = 1
(4w)0 = 1, for any w  0.
(−1)80 = (−1)1 = −1
−80 is read “the opposite of 80” and is equivalent to (−1)80: −80 = (−1)80 = (−1)1 = −1

13. The POWER Rule For any number a and any whole numbers m and n
In other Words: To RAISE a POWER to a POWER, MULTIPLY the exponents and leave the base unchanged

14. Quick Test of Power Rule

15. Example  Power Rule Simplify: a) (x3)4 b) (42)8
Solution a) (x3)4 = x34
= x12
Solution b) (42)8 = 428
= 416
Base is x
Base is 4

16. Raising a Product to a Power
For any numbers a and b and any whole number n,
In other Words: To RAISE A PRODUCT to a POWER, RAISE Each Factor to that POWER

17. Quick Test of Product to Power

18. Example  Product to Power
Simplify: a) (3x) b) (−2x3) c) (a2b3)7(a4b5)
Solutions
(3x)4 = 34x4 = 81x4
(−2x3)2 = (−2)2(x3)2 = (−1)2(2)2(x3)2 = 4x6
(a2b3)7(a4b5) = (a2)7(b3)7a4b5
= a14b21a4b Multiplying exponents
= a18b26 Adding exponents

19. Raising a Quotient to a Power
For any real numbers a and b, b ≠ 0, and any whole number n
In other Words: To Raise a Quotient to a power, raise BOTH the numerator & denominator to the power

20. Quick Test of Quotient to Power

21. Example  Quotient to a Power
Simplify: a) b) c)
Solution a)
Solution b)
Solution c)

22. Negative Exponents
Integers as Negative Exponents

23. Negative Exponents
For any real number a that is nonzero and any integer n
The numbers a−n and an are thus RECIPROCALS of each other

24. Example  Negative Exponents
Express using POSITIVE exponents, and, if possible, simplify.
a) m–5 b) 5–2 c) (−4)−2 d) xy–1
SOLUTION
a) m–5 =
b) 5–2 =

25. Example  Negative Exponents
Express using POSITIVE exponents, and, if possible, simplify.
a) m–5 b) 5–2 c) (−4)−2 d) xy−1
SOLUTION
c) (−4)−2 =
d) xy–1 =
Remember PEMDAS

26. More Examples
Simplify. Do NOT use NEGATIVE exponents in the answer. a) b) (x4)3 c) (3a2b4)3 d) e) f)
Solution a)

27. More Examples Solution b) (x−4)−3 = x(−4)(−3) = x12
c) (3a2b−4)3 = 33(a2)3(b−4)3
= 27 a6b−12 = d) e) f)

28. Factors & Negative Exponents
For any nonzero real numbers a and b and any integers m and n
A factor can be moved to the other side of the fraction bar if the sign of the exponent is changed

29. Examples  Flippers Simplify SOLUTION
We can move the negative factors to the other side of the fraction bar if we change the sign of each exponent.

30. Reciprocals & Negative Exponents
For any nonzero real numbers a and b and any integer n
Need to make a TEST for this
Any base to a power is equal to the reciprocal of the base raised to the opposite power

31. Examples  Flippers
Simplify
SOLUTION

32. Summary – Exponent Properties
1 as an exponent
a1 = a
0 as an exponent
a0 = 1
Negative Exponents (flippers)
The Product Rule
The Quotient Rule
The Power Rule
(am)n = amn
The Product to a Power Rule
(ab)n = anbn
The Quotient to a Power Rule
This summary assumes that no denominators are 0 and that 00 is not considered. For any integers m and n

33. WhiteBoard Work Problems From §1.6 Exercise Set
14, 24, 52, 70, 84, 92, 112, 130
Base & Exponent → Which is Which?

34. Astronomical Unit (AU)
All Done for Today
Astronomical Unit
(AU)

35. Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu
Chabot Mathematics
Appendix
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer –