1. Integer Exponents and Scientific Notation
5.1
Integer Exponents and Scientific Notation
Definition and Rules for Exponents
For all integers m and n and all real numbers a and b, the following rules apply.
Product Rule
Quotient Rule
Zero Exponent

2. Definition and Rules for Exponents (continued)
Negative Exponent
Power Rules
Special Rules

3. CAUTION A negative exponent does not indicate a negative number; negative exponents lead to reciprocals. For example,

4. EXAMPLE 1
Evaluate. a. b. a. b.

5. EXAMPLE 2
Write with only positive exponents and then evaluate.

6. EXAMPLE 3
Simplify so that no negative exponents are in the final result. Assume that all variables represent nonzero real numbers. a. b. c.

7. continued d.
Combination of rules

8. Scientific Notation
A number is written in scientific notation when it is expressed in the form
a × 10n
where 1 ≤ |a| < 10 and n is an integer.

9. Converting to Scientific Notation
Step 1 Position the decimal point. Place a caret, ^, to the right of the first nonzero digit, where the decimal point will be placed.
Step 2 Determine the numeral for the exponent. Count the number of digits from the decimal point to the caret. This number gives the absolute value of the exponent on 10.
Step 3 Determine the sign for the exponent. Decide whether multiplying by 10n should make the result of Step 1 greater or less. The exponent should be positive to make the result greater; it should be negative to make the result less.

10. EXAMPLE 4 . Write the number in scientific notation. a. 29,800,000
Step 1 Place a caret to the right of the 2 (the first nonzero digit) to mark the new location of the decimal point.
Step 2 Count from the decimal point, which is understood to be after the caret. .
29,800,000 = 2.9,800,000
Decimal point
Step 3 Since 2.98 is to be made greater, the exponent on 10 is positive.
29,800,000 = 2.98 × 107

11. continued Write the number in scientific notation. b. 0.0000000503
Step 1 Place a caret to the right of the 5 (the first nonzero digit) to mark the new location of the decimal point.
Step 2 Count from the decimal point 8 places, which is understood to be after the caret. =
Step 3 Since 5.03 is to be made less, the exponent 10 is negative.
= 5.03 × 10-8

12. Converting from Scientific Notation
Multiplying a number by a positive power of 10 makes the number greater, so move the decimal point to the right if n is positive in 10n.
Multiplying a number by a negative power of 10 makes the number less, so move the decimal point to the left if n is negative.
If n is 0, leave the decimal point where it is.

13. EXAMPLE 5 Write each number in standard notation. a. 2.51 ×103
= = 2510
Move the decimal 3 places to the right.
b. –6.8 ×10-5
= – = –
Move the decimal 4 places to the left.

14. Covert to Scientific Notation
Calculate using Scientific Notation
Covert to Scientific Notation
Rearrange / Regroup the numbers verses the exponents
Simplify the exponents
Calculate
Write in standard form

15. EXAMPLE 6
Evaluate

16. EXAMPLE 7
The distance to the sun is 9.3 × 107 mi. How long would it take a rocket traveling at 3.2 × 103 mph to reach the sun?
d = rt, so
It would take approximately 2.9 ×104.