1. Section 1.3
Integer Exponents

2. Objectives Bases and Positive Exponents Zero and Negative Exponents
Product, Quotient, and Power Rules
Order of Operations
Scientific Notation

3. Bases and Positive Exponents
The expression 82 is an exponential expression with base 8 and exponent 2.
Exponent
Base

4. Example
Using the given base, write each number as an exponential expression. a. 100,000 (base 10) b. 128 (base 2) Solution a. 100,000 b. 128

5. INTEGER EXPONENTS
Let a and b be nonzero real numbers and m and n be positive integers. Then 1. 2. 3.

6. Example
Simplify each expression. a. b. c. d. Solution a. b. c. d.

7. For any number a and integers m and n,
THE PRODUCT RULE
For any number a and integers m and n,

8. Example
Multiply and simplify. Use positive exponents. a. b. c. d. Solution a. b. c. d.

9. For any nonzero number a and integers m and n,
THE QUOTIENT RULE
For any nonzero number a and integers m and n,

10. Example
Simplify each expression. Use positive exponents. a. b. c. d. Solution a. b. c. d.

11. For any real number a and integers m and n,
RAISING POWERS TO POWERS
For any real number a and integers m and n,

12. Example
Simplify each expression. Use positive exponents. a. b. c. d.

13. For any real numbers a and b and integer n,
RAISING PRODUCTS TO POWERS
For any real numbers a and b and integer n,

14. Example
Simplify each expression. Use positive exponents. a. b. c. d.

15. For nonzero numbers a and b and any integer n,
RAISING QUOTIENTS TO POWERS
For nonzero numbers a and b and any integer n,

16. Example
Simplify each expression. Use positive exponents. a. b. c. d.

17. ORDER OF OPERATIONS
Using the following order of operations, first perform all calculations within parentheses and absolute values, or above and below the fraction bar. Then use the same order of operations to perform the remaining calculations.
1. Evaluate all exponential expressions. Do any negations after evaluating exponents.
2. Do all multiplication and division from left to right.
3. Do all addition and subtraction from left to right.

18. Example
Evaluate each expression. a. b.

19. SCIENTIFIC NOTATION
A real number a is in scientific notation when a is written as b  10n , where 1 ≤ |b| < 10 and n is an integer.

20. 3. If the decimal point was moved to the left, then a = b  10n.
WRITING A POSITIVE NUMBER IN SCIENTIFIC NOTATION
1. Move the decimal point in a number a until it represents a number b such that 1 ≤ b < 10.
2. Count the number of decimal places that the decimal point was moved. Let this positive integer be n. (If the decimal point is not moved, then a = a  100.)
3. If the decimal point was moved to the left, then a = b  10n.
If the decimal point was moved to the right, then a = b  10-n.

21. Important Powers of 10 Number 10-3 10-2 10-1 103 106 109 1012 Value
Thousandth
Hundredth
Tenth
Thousand
Million
Billion
Trillion

22. Example
Write each number in scientific notation. a. 475,000 b Solution a. 475,000 b.
Move the decimal point 6 places to the right.
Move the decimal point 5 places to the left.

23. Example Write each number in standard form. a. b. Solution
Move the decimal point 6 places to the right since the exponent is positive.
Move the decimal point 3 places to the left since the exponent is negative.