1. Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Section 5.7
Negative
Exponents and
Scientific
Notation
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1

3. Negative Exponents in Numerators and Denominators
If b is any real number other than 0 and n is a natural number, then
When a negative number appears as an exponent, switch the position of the base (from numerator to denominator or denominator to numerator) and make the exponent positive. The sign of the base does not change.

4. Negative Exponents
EXAMPLES

5. Negative Exponents

6. Objective #1: Example

7. Objective #1: Example

8. Objective #1: Example

9. Objective #1: Example

10. Objective #1: Example

11. Objective #1: Example

12. Objective #1: Example

13. Objective #1: Example

14. Objective #1: Example

15. Objective #1: Example

17. Simplifying Exponential Expressions
An exponential expression is “simplified” when
a. Each base occurs only once.
b. No parentheses appear.
c. No powers are raised to powers.
d. No negative or zero exponents appear.

18. Simplification Techniques
Simplifying Exponential Expressions
Simplification Techniques
Examples
If necessary, remove parentheses by using the Products to Powers Rule or the Quotient to Powers Rule.
If necessary, simplify powers to powers by using the Power Rule.

19. Simplification Techniques
Simplifying Exponential Expressions
Simplification Techniques
Examples
Be sure each base appears only once in the final form by using the Product Rule or Quotient Rule
If necessary, rewrite exponential expressions with zero powers as 1. Furthermore, write the answer with positive exponents by using the Negative Exponent Rule

20. Simplifying Exponential Expressions
EXAMPLE
Cube each factor in the numerator.
Multiply powers using (bm)n = bmn.
Division with the same base, subtract exponents.
When a negative number appears as an exponent, switch the position of the base. The x-2 moves from numerator to denominator as x2.

21. Simplifying Exponential Expressions
EXAMPLE
Cube each factor in the numerator.
Multiply powers using (bm)n = bmn.
Division with the same base, subtract exponents.
When a negative number appears as an exponent, switch the position of the base. The x-2 moves from numerator to denominator as x2.

22. Objective #2: Example

23. Objective #2: Example

24. Objective #2: Example

25. Objective #2: Example

26. Objective #2: Example

27. Objective #2: Example

28. Objective #2: Example

29. Objective #2: Example

31. Scientific Notation Scientific Notation
At times you may find it necessary to work with really large numbers, or alternately, really small numbers. Here you will learn how to write these often cumbersome numbers in scientific notation.
Scientific Notation
A positive number is written in scientific notation when it is expressed in the form
Where a is a number greater than or equal to 1 and less than 10
and n is an integer.

32. Converting from Scientific to Decimal Notation
Scientific Notation
Converting from Scientific to Decimal Notation
We can use n, the exponent on the 10 in , to change a number in scientific notation to decimal notation. If n is positive, move the decimal point in a to the right n places. If n is negative, move the decimal point in a to the left places.

33. Objective #3: Example

34. Objective #3: Example

36. Converting from Decimal to Scientific Notation
Determine a, the numerical factor. Move the decimal point in the given number to obtain a number greater than or equal to 1 and less than 10.
Determine n, the exponent on 10n. The absolute value of n is the number of places the decimal was moved. The exponent n is positive if the given number is great than 10 and negative if the given number is between 0 and 1.

37. Objective #4: Example

38. Objective #4: Example

40. Computations with Scientific Notation

41. Computations with Scientific Notation
EXAMPLE
Perform the indicated computation, writing the answer in scientific notation.
SOLUTION
Regroup factors
Multiply
Simplify
Rewrite in scientific notation

42. Computations with Scientific Notation
EXAMPLE
Perform the indicated computation, writing the answer in scientific notation.
SOLUTION
Group factors
Divide
Simplify
Write in scientific notation

43. Objective #5: Example

44. Objective #5: Example

45. Objective #5: Example

46. Objective #5: Example

47. Objective #5: Example

48. Objective #5: Example

50. Scientific Notation
EXAMPLE
The area of Alaska is approximately acres. The state was purchased in 1867 from Russia for $7.2 million. What price per acre, to the nearest cent, did the United States pay Russia?
SOLUTION
We will first write 7.2 million in scientific notation. It is:
Now we can answer the question.

51. Scientific Notation
EXAMPLE
The area of Alaska is approximately acres. The state was purchased in 1867 from Russia for $7.2 million. What price per acre, to the nearest cent, did the United States pay Russia?
SOLUTION
We will first write 7.2 million in scientific notation. It is:
Now we can answer the question.

52. Scientific Notation Divide ‘dollar amount’ by ‘acreage’ Simplify
CONTINUED
Divide ‘dollar amount’ by ‘acreage’
Simplify
Divide
Subtract
Therefore, since dollars/acre equals $0.0197/acre, then the price per acre for Alaska was approximately 2 cents per acre.

53. Scientific Notation Divide ‘dollar amount’ by ‘acreage’ Simplify
CONTINUED
Divide ‘dollar amount’ by ‘acreage’
Simplify
Divide
Subtract
Therefore, since dollars/acre equals $0.0197/acre, then the price per acre for Alaska was approximately 2 cents per acre.

54. Objective #6: Example

55. Objective #6: Example