1. Exponents and Polynomials
Chapter 3
Exponents and Polynomials

2. 10.1
Exponents

3. Exponents
Exponents that are natural numbers are shorthand notation for repeating factors.
34 = 3 • 3 • 3 • 3
3 is the base
4 is the exponent (also called power)
Note by the order of operations that exponents are calculated before other operations.

4. Evaluating Exponential Expressions
Example
Evaluate each of the following expressions. 34
= 3 • 3 • 3 • 3
= 81
(–5)2
= (– 5)(–5)
= 25
–62
= – (6)(6)
= –36
(2 • 4)3
= (2 • 4)(2 • 4)(2 • 4)
= 8 • 8 • 8
= 512
3 • 42
= 3 • 4 • 4
= 48

5. Evaluating Exponential Expressions
Example
Evaluate each of the following expressions.
a.) Find 3x2 when x = 5.
3x2 = 3(5)2
= 3(5 · 5)
= 3 · 25
= 75
b.) Find –2x2 when x = –1.
–2x2 = –2(–1)2
= –2(–1)(–1)
= –2(1)
= –2

6. The Product Rule
If m and n are positive integers and a is a real number, then
am · an = am+n
For example,
32 · 34
= 32+4
= 36
x4 · x5
= x4+5
= x9
z3 · z2 · z5
= z3+2+5
= z10
(3y2)(– 4y4)
= 3 · y2 (– 4) · y4
= 3(– 4)(y2 · y4)
= – 12y6

7. Helpful Hint Don’t forget that 35 ∙ 37 = 912
Add exponents.
35 ∙ 37 = 912
Common base not kept.
35 ∙ 37 = 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3
5 factors of 3.
7 factors of 3.
= 312
12 factors of 3, not 9.
In other words, to multiply two exponential expressions with the same base, we keep the base and add the exponents. We call this simplifying the exponential expression.

8. Helpful Hint
Don’t forget that if no exponent is written, it is assumed to be 1.

9. The Power Rule
If m and n are positive integers and a is a real number, then
(am)n = amn
For example,
(23)3
= 23·3
= 29
(x4)2
= x4·2
= x8

10. The Power of a Product Rule
If n is a positive integer and a and b are real numbers, then
(ab)n = an · bn
For example,
(5x2y)3
= 53 · (x2)3 · y3
= 125x6 y3

11. The Power of a Quotient Rule
If n is a positive integer and a and c are real numbers, then
For example,

12. The Quotient Rule
If m and n are positive integers and a is a real number, then
For example,
Group common bases together.

13. Zero Exponent a0 = 1, a ≠ 0 Note: 00 is undefined. For example, 50 = 1
(xyz3)0
= x0 · y0 · (z3)0
= 1 · 1 · 1 = 1
–x0
= –(x0)
= – 1