section 1.6 natural number exponents and order of operations

Section 1.6

Natural Number Exponents and Order of Operations

Objective 1: Use natural number exponents.

2 2 2 2 2 2 2 2

82 2 2 2 2 2 2 2 2

1.6 Natural Number Exponents and Order of Operations

Repeated Multiplication: Exponential notation is a concise notation to indicate repeatedly multiplying the same factor a given number of times. The expression

can be written using exponentialnotation. The base, or factor being repeatedly multiplied, is 2. The exponent, or number of times the factor is repeated, is 8. So Exponential notation is a concise notation to indicate repeatedly multiplying the same factor a given number of times.

.

Algebraically

Verbally Numerical

Examples

For any natural number n,

with base b and exponent n.

For any natural number n, is the product of b used as a factor n times. The expression is read as “b to the nth power.”

factors of

n

n b

b b b b

nb

35

24

Exponential Notation

1.

3.

2.

4.

4 4 4 4 4 x x x

42 3

5x

Write each expression in exponential form.

Write each expression in expanded form.

To avoid errors it is very important to identify the base of an exponential expression. If an exponent is on a number or variable, then that number or variable is the base. If an exponent is outside a pair of grouping symbols, then the contents of this pair of grouping symbols is the base. Identify the correct base, exponent and expanded form of each expression.

Exponential Expression Base Exponent Expanded Form

5.

6.

7.

8.

24

24

43x

43x

9. 10. 11.26 624

23

Mentally evaluate each expression.

12. 13. 14.

Mentally evaluate each expression.

25 25 21

1

16. and have the same value although the bases are different. Can you explain this?

15. Can you explain the subtle distinction between

and ?

43 43

35 35

17.(a) Which of the following do you think is the correct evaluation of the expression ? 6 3 3 7 4

Option I Option II Option III

6 3 3 7 4 9 3 7 4

6 7 4

1 4

4

6 3 3 7 4 6 1 28

7 28

21

6 3 3 7 4 9 4 4

94

49

(b) Try evaluating the above expression on your calculator. Do you agree with the calculator result?

Objective 2: Use the standard order of operations.

A major objective in this section is to master the order of operations. The order of operations gives a consistent method for evaluating mathematical expressions like the one above.

Standard Order of Operations

Step 1: Start with the expression within the innermost pair of ________________ ____________________.

Step 2: Perform all __________________.

Step 3: Perform all __________________ and __________________ as they appear from left to right.

Step 4: Perform all __________________ and __________________ as they appear from left to right.

. The first four grouping symbols contain both a beginning symbol before and an ending symbol after the group. In the radical symbol and the fraction bar the group is determined by the length of the horizontal bar.

Some common grouping symbols are , , , , , and

18.

19. 20. 23 80 81

Use the standard order of operations to perform the indicated operations. Use a calculator only to check your results.

21.

22. 23.

Use the standard order of operations to perform the indicated operations. Use a calculator only to check your results.

81 12 4 3 12 4 3

Use the standard order of operations to perform the indicated operations. Use a calculator only to check your results.

12 4 3 12 4 3 24.

25.

26.

27.

Use the standard order of operations to perform the indicated operations. Use a calculator only to check your results.

2 27 5 27 5

28.

29.

Use the standard order of operations to perform the indicated operations. Use a calculator only to check your results.

28 20 4 6 28 20 4 6

30.

31.

Use the standard order of operations to perform the indicated operations. Use a calculator only to check your results.

4 3 3 4

2

36 9 3 2

32.

33.

Use the standard order of operations to perform the indicated operations. Use a calculator only to check your results.

27 7 4 13 2 2

2 2

5 4

5 4

34.

35.

Use the standard order of operations to perform the indicated operations. Use a calculator only to check your results.

5 3 4 6 5 1 5 8 6 9

36.

37.

Use the standard order of operations to perform the indicated operations. Use a calculator only to check your results.

4 25 4 1 4

2

2 23 420 2 9 6

Objective 3: Use the distributive property of multiplication over addition.

The distributive property can be used in two ways: to expand expressions that are in a factored form and to factor expressions with terms that have a common factor. Distributive Property of Multiplication Over Addition

Algebraically Verbally Numerical Examples

For all real numbers a, b, and c,

and

.

Multiplication distributes over addition. a b c ab ac

b c a ba ca

3 4 5 3 4 3 5

4 5 3 4 3 5 3

Use the distributive property to expand each expression in the first column and to factor each expression in the second column.

38.

39. 6 3 5x 2 10x

Use the distributive property to expand each expression in the first column and to factor each expression in the second column.

40.

41. 18 24x y 4 7 5x y

42.

43.

Use the distributive property to expand each expression in the first column and to factor each expression in the second column.

2x 1 2 3x

a b c b c a

a b c a c b

a b c ab ac

44. What property justifies the fact that ?

46. What property justifies the fact that ?

45. What property justifies the fact that ?

Adding Like Terms: Use the distributive property to combinelike terms.

47.

48.7 4x x 8 6x x

49.

50.3 7 11x y x 3 13x y x

Adding Like Terms: Use the distributive property to combinelike terms.

51.

52. (3 5 ) (2 7 )x y x y (2 7 ) (3 4 )x y x y

Adding Like Terms: Use the distributive property to combinelike terms.

53.

54.4( 3) 5(3 7)x x 8( 3) 2(2 1)x x

Adding Like Terms: Use the distributive property to combinelike terms.

Phrases Used To Indicate Exponentiation

Key Phrases Verbal Examples Algebraic Examples

To a power "3 to the 6th power"

Raised to "y raised to the 5th power"

Squared "4 squared"

Cubed "x cubed"

63

5y

24

3x

55. x raised to the fourth power

56. The square of the quantity x plus two.

Translate each verbal statement into algebraic form.

57. 58.12 84 9

210 53

Write each expression in the horizontal one-line format used by calculators.

Each expression is given in the horizontal one-line format used by calculators. Rewrite each expression in the standard algebraic format.

59. 60.^ 2 4 / 2x x ^ 2 4 / 2x x