1. Section 1.5 Multiplication of Real Numbers

2. 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.

3. Notations for the Product of the Factors x and y:

4. Phrases Used To Indicate Multiplication: Key PhraseVerbal ExampleAlgebraic 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”

5. 1. Translate each verbal statement into algebraic form. a times six

6. 2. Translate each verbal statement into algebraic form. x is multiplied by four

7. 3. Translate each verbal statement into algebraic form. The product of p and q

8. 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. AlgebraicallyNumerical Examples

9. 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. SumSignProductSign (positive)+(positive)=(positive)●(positive)= (positive)+(negative)=(positive)●(negative)= (negative)+(positive) =(negative)●(positive) = (negative)+(negative)=(negative)●(negative)= (0)+(positive)=(0)●(positive)= (0)+(negative)=(0)●(negative)=

10. A Factor of −1 Algebraically For any real number,. Verbally The product of negative one and any real number is the opposite of that real number. Numerical Examples and

11. 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: ____________ 5.

12. Sign: ______ Value: ____________ 6. 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.

13. Sign: ______ Value: ____________ 7. 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.

14. Sign: ______ Value: ____________ 8. and, where 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 as.

15. Sign: ______ Value: ____________ 9., where,, and 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 as.

16. Properties of radicals will be studied in more detail in Chapter 9. For now, use the fact that for both and to evaluate each expression. 10. 11.

17. 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.

18. Algebraically Verbally Numerical Example Multiplying Fractions: To multiply two fractions, multiply the numerators and multiply the denominators. forand

19. 14. Adding Fractions vs. Multiplying Fractions: Add the fractions in the first column and multiply the fractions in the second column. AddingMultiplying (a)(b) (c)(d)

20. 14. Adding Fractions vs. Multiplying Fractions: (e) What is absolutely necessary when adding two fractions? (f) Is this also necessary when multiplying fractions?

21. 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 Zero has no multiplicative inverse. is one., and

22. NumberMultiplicative 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.

23. 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?

24. Algebraically Verbally Numerical Example Commutative Property of Multiplication Objective: Use the commutative and associative properties of multiplication. The product of two factors in either order is the same.

25. Algebraically Verbally Numerical Example Associative Property of Multiplication Factors can be regrouped without changing the product.

26. 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 Addition II. Commutative Property of Multiplication III. Associative Property of Addition IV. Associative Property of Multiplication 22. _____ 24. _____ 23. _____ 25. _____ 26. _____ 27. _____

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?

28. 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 = 2 8. Exponential notation is a nice way to indicate repeatedly multiplying the same factor a given number of times. Objective: Use natural number exponents.

29. Exponential Notation: Verbally Numerical Examples Algebraically For any natural number n, with base b and exponent n. For any natural number n, as a factor n times. The expression is the product of b used “b to the nth power.” is read as

30. Phrases Used To Indicate Exponentiation: Key PhraseVerbal ExampleAlgebraic Example To a power"3 to the 6th power" Raised to"y raised to the 5th power" Squared"4 squared" Cubed"x cubed"

31. Translate each verbal statement into algebraic form. x raised to the fourth power29.

32. Translate each verbal statement into algebraic form. y squared30.

33. Rewrite each product below using exponential notation, and use a calculator to compute the value. ProductExponential Notation Value Example: 31. 32. 33.

34. Rewrite each product below using exponential notation, and use a calculator to compute the value. ValueExponential Notation Product in factored form Example: 125 34. −7776 35. 64 36. 81

35. 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.

36. ExpressionBaseExponentValue 37.42 38. 39. 40. Identify the correct base, exponent and value of each expression.

37. ExpressionBaseExponentValue 41. 4 42. 43. 44. Identify the correct base, exponent and value of each expression.

38. To evaluate the expression 3 5 on your calculator, you may use repeated multiplication, or you may want to use the exponent key.

39. 45. Can you explain the subtle distinction between and ?

40. 46. andhave the same value although the bases are different. Can you explain this?

41. The following expressions are often misinterpreted. Pay careful attention to the base and write each exponential expression in expanded form. 47. 48.

42. 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, 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.

43. Find the area of each triangle. A = ______49. 4 cm 7 cm

44. Find the area of each triangle. A = ______50. 3 in 8 in

45. The formula for Fahrenheit temperature is given by 51.. Find the Fahrenheit temperature if the Celsius temperature C is

46. 52. The formula for the perimeter of a rectangle is given by. Find the perimeter P of a rectangle if the length l is 20 meters and the width w is 35 meters.

47. 53. The formula for the amount in a bank account paying a simple interest rate R for T years is given by 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.,