Section 1.5

Multiplication of Real Numbers

1.5 Lecture Guide: Multiplication of Real Numbers and Natural Number Exponents

Objective: Multiply positive and negative real numbers. Although memorization is not generally the best way to learn mathematical concepts, it is very helpful to have the following key facts memorized when performing multiplication and addition.

Notations for the Product of the Factors x and y:

xy x y x y x y x y x y

Phrases Used To Indicate Multiplication:Key Phrase Verbal Example Algebraic Example

Times "x times y"

Produce "The product of 5 and 7"

Multiplied by "The rate r is multiplied by the time t"

Twenty percent of

"Twenty percent of x"

Twice "Twice y"

Double “Double the price P”

Triple “Triple the coupon value V”

0.20x

rt

5 7

xy

2y

2P3V

1.

Translate each verbal statement into algebraic form.

a times six

2.

Translate each verbal statement into algebraic form.

x is multiplied by four

3.

Translate each verbal statement into algebraic form.

The product of p and q

Multiplication of Two Real Numbers:

Like signs: Multiply the absolute values of the two factors and use a positive sign for the product.

Unlike signs: Multiply the absolute values of the two factors and use a negative sign for the product.

Zero factor: The product of 0 and any other factor is 0.

Algebraically Numerical Examples

3 6 ______ 3 6 ______

3 6 ______

3 6 ______

3 0 ______

0 6 ______

The Sign of a Sum vs. the Sign of a Product: 4. Fill in the correct sign of the sum or product below. If the sign cannot be determined, give an explanation.

Sum Sign Product Sign

(positive)+(positive)= (positive)●(positive)=

(positive)+(negative)= (positive)●(negative)=

(negative)+(positive) = (negative)●(positive) =

(negative)+(negative)= (negative)●(negative)=

(0)+(positive)= (0)●(positive)=

(0)+(negative)= (0)●(negative)=

A Factor of −1

AlgebraicallyFor any real number a, 1 a a .

VerballyThe product of negative one and any real number is the opposite of that real number.

Numerical Examples1 3

and

1 4

Give the sign of the following product by visually counting the number of negative factors and then, if needed, use a calculator to determine the products.

2 3 8 5 6 1

Sign: ______

Value: ____________

5.

Sign: ______

Value: ____________

6. 1 2 3 4 1 5

Give the sign of the following product by visually counting the number of negative factors and then, if needed, use a calculator to determine the products.

Sign: ______

Value: ____________

7. 4 2 0 9

Give the sign of the following product by visually counting the number of negative factors and then, if needed, use a calculator to determine the products.

Sign: ______

Value: ____________

8. 2y and 3x , where xy

Evaluate each expression. Practice using the store feature on your calculator to check your results. When evaluating expressions involving variables, it is important to think of x as 1 x .

Sign: ______

Value: ____________

9. abc , where 1a , 3b , and 7c

Evaluate each expression. Practice using the store feature on your calculator to check your results. When evaluating expressions involving variables, it is important to think of x as 1 x .

Properties of radicals will be studied in more detail in Chapter 9. For now, use the fact that a b ab for both 0a and 0b to evaluate each expression.

10. 2 8

11. 6 24

Product of Negative Factors:

12. The product is ____________ if the number of negative factors is even.

13. The product is ____________ if the number of negative factors is odd.

Algebraically

Verbally

Numerical Example

Multiplying Fractions:

To multiply two fractions, multiply the numerators and multiply the denominators.

for 0b and 0d a c acb d bd

2 37 5

14. Adding Fractions vs. Multiplying Fractions: Add the fractions in the first column and multiply the fractions in the second column.

Adding Multiplying

1 512 12

1 512 12

(a) (b)

(c) (d)2 13 4

2 13 4

14. Adding Fractions vs. Multiplying Fractions:

(e) What is absolutely necessary when adding two fractions?

(f) Is this also necessary when multiplying fractions?

Algebraically

Verbally

Numerical Example

Reciprocals or Multiplicative Inverses:

For any real number a other than zero, the product of

the number a and its multiplicative inverse

For any real number 0a 1

1aa

1a

Zero has no multiplicative inverse.

is one.

14 1

4

, and4 3

13 4

Number Multiplicative Inverse

Product

Example:

15.

16.

17.

18.

Give the multiplicative inverse of each of the following real numbers and then multiply the number by its multiplicative inverse.

7

5

4

4

74

17

17 1

7

15

19. Is the multiplicative inverse of a positive number positive or negative?

20.

21. Does every real number have a multiplicative inverse?

Is the multiplicative inverse of a negative number positive or negative?

Algebraically

Verbally

Numerical Example

Commutative Property of Multiplication

Objective: Use the commutative and associative properties of multiplication.

ab ba

The product of two factors in either order is the same.

5 6

6 5

Algebraically

Verbally

Numerical Example

Associative Property of Multiplication

ab c a bc

Factors can be regrouped without changing the product.

2 3 4

2 3 4

Identify the property used to justify the equality of the two expressions in each equation below. Select from the following list:

I. Commutative Property of AdditionII. Commutative Property of MultiplicationIII. Associative Property of AdditionIV. Associative Property of Multiplication

22. _____ 3 4( 2) 3 4 2x x

24. _____ 3 2 3 2x x

23. _____ 2 3 2 3x x x x

25. _____

5 4 4 5x x

2 5 2 5x x

26. _____ 3 5 5 3x x x x 27. _____

28. Think about the difference in the terms “regroup” and “reorder”. (a) Which term applies to the commutative property?

(b) Which term applies to the associative property?

Repeated Multiplication: The expression 2●2●2●2●2●2●2●2 can be written using exponential notation. The base, or factor being repeatedly multiplied, is 2. The exponent, or number of times the factor is repeated, is 8. So 2●2●2●2●2●2●2●2 = 28. Exponential notation is a nice way to indicate repeatedly multiplying the same factor a given number of times.

Objective: Use natural number exponents.

Exponential Notation:

Verbally

Numerical Examples

Algebraically For any natural number n,

factors of

n

n b

b b b b with base b and exponent n.

For any natural number n, nbas a factor n times. The expression

is the product of b usednb

“b to the nth power.” is read as

35 24

Phrases Used To Indicate Exponentiation:

Key Phrase Verbal Example Algebraic Example

To a power "3 to the 6th power"

Raised to "y raised to the 5th power"

Squared "4 squared"

Cubed "x cubed"

635y

243x

Translate each verbal statement into algebraic form.

x raised to the fourth power 29.

Translate each verbal statement into algebraic form.

y squared 30.

Rewrite each product below using exponential notation, and use a calculator to compute the value.

Product Exponential Notation

Value

Example:

31.

32.

33.

4 4 4 4 4

7 7 7

9 9

4 4 4 4

54 1024

Rewrite each product below using exponential notation, and use a calculator to compute the value.

Value Exponential Notation

Product in factored form

Example:

125

34. −7776

35. 64

36. 81

5 5 5 35

56

62

43

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.

Expression Base Exponent Value

37. 4 2

38.

39.

40.

24

35

40.2

3

2

3

Identify the correct base, exponent and value of each expression.

Expression Base Exponent Value

41. 4

42.

43.

44.

Identify the correct base, exponent and value of each expression.

42 2

23 1

30.1

33

4

To evaluate the expression 35 on your calculator, you may use repeated multiplication, or you may want to use the exponent key.

45. Can you explain the subtle distinction between

43 and 43 ?

46. 35 and 35 have the same value although the bases

are different. Can you explain this?

The following expressions are often misinterpreted. Pay careful attention to the base and write each exponential expression in expanded form.

47. 24x

48. 24x

Objective: Use algebraic formulas. Algebraic formulas are used in nearly all areas of math and science. A formula describes a relationship between specific variables. For example, the area A of a triangle is given by

the formula 12

A b h , where b represents the length of the

base of the triangle and h represents the height of the triangle. This relationship holds true for all triangles.

Find the area of each triangle.

A = ______49.

4 cm

7 cm

Find the area of each triangle.

A = ______50.

3 in

8 in

The formula for Fahrenheit temperature is given by 51. 9

325

F C . Find the Fahrenheit temperature if the

Celsius temperature C is 55

52. The formula for the perimeter of a rectangle is given by 2 2P l w . Find the perimeter P of a rectangle if the

length l is 20 meters and the width w is 35 meters.

53. The formula for the amount in a bank account paying a simple interest rate R for T years is given by A P PRT where P is the principal or initial amount. Find the amount in a bank account after 1 year if there was an initial deposit of $5,000 and the account earned 5% simple interest.

,