1. Chapter 4
Polynomials

2. Exponents and Their Properties
4.1
Multiplying Powers with Like Bases
Dividing Powers with Like Bases
Zero as an Exponent
Raising a Power to a Power
Raising a Product or a Quotient to a Power

3. The Product Rule
For any number a and any positive integers m and n, (To multiply powers with the same base, keep the base and add the exponents.)

4. Multiply and simplify each of the following
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) Solution a) x3  x5 = x3+5 Adding exponents = x8

5. (cont) b) 62  67  63 c) (x + y)6(x + y)9 d) (w3z4)(w3z7) Solution
= 612
c) (x + y)6(x + y)9 = (x + y)6+9
= (x + y)15
d) (w3z4)(w3z7) = w3z4w3z7
= w3w3z4z7
= w6z11

6. The Quotient Rule
For any nonzero number a and any positive integers m and n for which m > n, (To divide powers with the same base, subtract the exponent of the denominator from the exponent of the numerator.)

7. Divide and simplify each of the following
Divide and simplify each of the following. (Here “simplify” means express the quotient as one base to a power whenever possible.) a) b) c) d) Solution a) b)

8. (cont) c) d)

9. The Exponent Zero
For any real number a, with a ≠ 0, (Any nonzero number raised to the 0 power is 1.)

10. Simplify: a) 12450 b) (3)0 c) (4w)0
d) (1)80 e) 90.
Solution
a) = 1
b) (3)0 = 1
c) (4w)0 = 1, for any w  0.
d) (1)80 = (1)1 = 1
e) 90 is read “the opposite of 90” and is equivalent to (1)90: 90 = (1)90 = (1)1 = 1

11. The Power Rule
For any number a and any whole numbers m and n, (am)n = amn. (To raise a power to a power, multiply the exponents and leave the base unchanged.)

12. Simplify: a) (x3)4 b) (42)8
Solution
a) (x3)4 = x34
= x12
b) (42)8 = 428
= 416

13. Raising a Product to a Power
For any numbers a and b and any whole number n, (ab)n = anbn. (To raise a product to a power, raise each factor to that power.)

14. Simplify: a) (3x)4 b) (2x3)2 c) (a2b3)7(a4b5)
Solution
a) (3x)4 = 34x4
= 81x4
b) (2x3)2 = (2)2(x3)2
= 4x6
c) (a2b3)7(a4b5) = (a2)7(b3)7a4b5
= a14b21a4b Multiplying exponents
= a18b Adding exponents

15. Raising a Quotient to a Power
For any real numbers a and b, b ≠ 0, and any whole number n, (To raise a quotient to a power, raise the numerator to the power and divide by the denominator to the power.)

16. Simplify: a) b) c)
Solution a) b) c)

17. Definitions and Properties of Exponents
For any whole numbers m and n,
1 as an exponent:
a1 = a
0 as an exponent:
a0 = 1
The Product Rule:
The Quotient Rule:
The Power Rule:
(am)n = amn
Raising a product to a power:
(ab)n = anbn
Raising a quotient to a power: