1. Section 1.5 Homework questions?

2. Section Concepts 1.5 Multiplication and Division of Integers 1.Multiplication of Integers 2.Multiplying Many Factors 3.Exponential Expressions 4.Division of Integers

3. Section 1.5 Multiplying Integers The multiplication rules for integers are determined from a pattern of products. Consider the following problems: 2 3 = 2 2 = The pattern decreases 2 1 =by two with each row. 2 0 = 2 (-1) = The product of a 2 (-2) = positive number and a 2 (-3) = negative number is negative.

4. Section 1.5 Multiplying Integers Now start the pattern with a negative integer. -2 3 = The pattern increases -2 2 =by two with each row. -2 1 = -2 0 = The product of a -2 (-1) =negative number and a -2 (-2) =negative number is -2 (-3) = positive.

5. PROCEDUREMultiplying Signed Numbers 1.The product of two numbers with the same sign is positive. Examples: (5)(6) = 30 (-5)(-6) = 30 2.The product of two numbers with different signs is negative. Examples:4(-8) = -32 -4(8) = -32 3.The product of any number and zero is zero. Examples:5(0) = 0 0(-7) = 0

6. Example Multiply. 1 -8(-6) -5(10) (14)(-2) 7(9) -11(0)

7. Example -2(-6) 3(-10) -12(4) 8(5) -7(0) Multiply. 2You Try.

8. Section 1.5 Translating to a Mathematical Expression Operation Words for Multiplication Product Times Multiply Of By Twice (times 2)

9. Example 3Translate to a mathematical expression. Then simplify. The product of 7 and -8. -3 times -12 5 multiplied by -6

10. Example 4You Try – Translate to a mathematical expression. Then simplify. The product of -12 and -6 -4 times -11 Multiply 5 by -9

11. Example Multiply the factors from left to right. 5Multiplying Many Factors (-2)(-2) (-2)(-2)(-2) (-2)(-2)(-2)(-2) = 4 = 4(-2) = 4(-2)(-2) = -8 = -8(-2) = 16 These products indicate the following rules: The product of an even number of negative factors is positive The product of an odd number of negative factors is negative

12. Example 6Multiply. (-3)(-2)(-4)You Try (-5)(-2)(-7)(-1)(-3)(-4)(-8)(-1) (-4)2(-1)(5)(-1)(-4)(-6)(5)

13. PROCEDUREExponential Expressions “negative 4 squared” “negative 1 times 4²” (-4 ∙ -4) -1 ∙ (4)(4) = 16-1 ∙ 16 = -16

14. Example 7Simplify.

15. Example 8Simplify. You Try

16. DEFINITION Division of Integers The sign rules for dividing integers are the same as the sign rules for multiplying integers, because we can check our division answers by multiplying. 24 ÷ 6 = 4Check: 4 ∙ 6 = 24 -24 ÷ 6 = -4 Check: -4 ∙ 6 = -24 24 ÷ (-6) = -4Check:-4 ∙ (-6) = 24 -24 ÷ (-6) = 4Check: 4 ∙ (-6) = -24

17. PROCEDUREDividing Signed Numbers 1.The quotient of two numbers with the same sign is positive. 2.The quotient of two numbers with different signs is negative. 3.Zero divided by any nonzero number is zero. 4.Any nonzero number divided by zero is undefined.

18. Example 9Divide. 50 ÷ (-5) 0 ÷ (-8) -36 ÷ 3 -9 ÷ 0

19. Example 10Divide. You Try -40 ÷ 1018 ÷ -2 0 ÷ 3-4 ÷ 0

20. Example 11Translate to a mathematical expression. Then simplify. The quotient of 25 and -5 -14 divided by -7 4 divided into -24

21. Example 12You Try. Translate to a mathematical expression and simplify. The quotient of -100 and 10 -56 divided by -8 6 divided into -12

22. Example 13Applications During a 10-min period, a SCUBA diver’s depth changed by -60 ft. Find the average change in depth per minute. Stephanie wrote five checks to the employees of her business, each for $225. If the original balance in her checking account was $890, what is her new balance?

23. Example 14Applications. You Try When a severe winter storm moved through Albany, New York, the change in temperature between 4:00 pm and 7:00 pm was -27⁰F. What was the average hourly change in temperature? Sarah’s checking account had $320 when she wrote two checks for $150, and one check for $82. What is her new balance?

24. DEFINITIONMultiplying Signed Numbers 1.The product of two numbers with the same sign is positive. Examples: (5)(6) = 30 (-5)(-6) = 30 2.The product of two numbers with different signs is negative. Examples:4(-8) = -32 -4(8) = -32 3.The product of any number and zero is zero. Examples:5(0) = 0 0(-7) = 0

25. DEFINITIONDividing Signed Numbers 1.The quotient of two numbers with the same sign is positive. 2.The quotient of two numbers with different signs is negative. 3.Zero divided by any nonzero number is zero. 4.Any nonzero number divided by zero is undefined.