1. Objectives Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Dividing Whole Numbers; Square Roots; Solving Equations 1. Divide whole numbers. 2. Solve equations containing an unknown factor. 3. Solve applications involving division. 4. Find the square root of a perfect square. 5. Solve applications involving square roots. 1.4

2. Slide 2 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 1 Divide whole numbers.

3. Slide 3 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Definition Division: Repeated subtraction of the same number. When we write a division sentence, the dividend is divided by the divisor and the quotient is the answer. Notation: DividendDivisorQuotient 20 ÷ 5 = 4

4. Like multiplication there are several ways to indicate division. Slide 4 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slash: Fraction bar: Long division:

5. Slide 5 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Definition Remainder: The amount left over after dividing two whole numbers.

6. Let’s explore a few properties of division… Like subtraction, division is neither commutative nor associative… Slide 6 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Division is not commutative: Division is not associative:

7. What if the divisor is 1 as in… Slide 7 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley To check the quotient, we must be able to multiply the quotient by the divisor to get the dividend. We learned that the product of 1 and a number is the number, so…, which means Conclusion:

8. Slide 8 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley When any number is divided by 1, the quotient is the number (the dividend). Division Property In math language: …when n is any number.

9. What if the divisor is 0 as in ? Slide 9 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley To check the quotient, we must be able to multiply the quotient by the divisor to get the dividend. Because the product of any number and 0 is 0, the equation has no solution. So…has no numerical quotient. Conclusion:

10. Slide 10 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley If the divisor is 0 and the dividend is any number other than 0, the quotient is undefined. Division Property In math language: or is undefined, when n ≠ 0.

11. What if the dividend is 0 Slide 11 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley as in ? Again, we must be able to multiply the quotient by the divisor to get the dividend. Conclusion:

12. But consider this…0 divided by 0. Slide 12 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley To check we must be able to multiply the quotient by the divisor to get the dividend. Actually…the product of any number and 0 is 0, so we cannot determine a value for… We say that it is indeterminate.

13. When 0 is divided by any number other than 0, the quotient is 0. When 0 is divided by itself, the quotient is indeterminate. Slide 13 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Conclusion:

14. Slide 14 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley When 0 is divided by any number other than 0, the quotient is 0. Division Properties In math language: when n ≠ 0. In math language: oris indeterminate.

15. Slide 15 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley When a number (other than 0) is divided by itself, the quotient is 1. Division Property In math language: when n ≠ 0.

16. Slide 16 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 1 Determine the quotient and explain your answer. a. 34 ÷ 1 = b. 27 ÷ 0 = Property?

17. Slide 17 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 1 Determine the quotient and explain your answer c. 0 ÷ 16 b. 45 ÷ 45 Property?

18. Slide 18 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2 is an exact divisor for all even numbers. Even numbers have 0, 2, 4, 6, or 8 in the ones place. Divisibility Rules Example: 3596 is divisible by 2 because it is an even number. 3596 has the digit 6 in the ones place.

19. Slide 19 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3 is an exact divisor for a number if the sum of all digits in the number is divisible by 3. Divisibility Rules Example: 58,014 is divisible by 3 because the sum of its digits is …. 5 + 8 + 0 + 1 + 4 = 18, which is divisible by 3.

20. Slide 20 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5 is an exact divisor for numbers that have 0 or 5 in the ones place. Divisibility Rules Example: 91,285 is divisible by 5 because it has a 5 in the ones place.

21. Slide 21 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 2 Use divisibility rules to determine whether the given number is divisible by 2. a. 45,091 b. 691,134

22. Slide 22 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 3 Use divisibility rules to determine whether the given number is divisible by 3. a. 76,413 b. 4256

23. Slide 23 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 4 Use divisibility rules to determine whether the given number is divisible by 5. a. 2,380,956 b. 49,360

24. Slide 24 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 6 Divide. 42,017 ÷ 41 Solution:

25. Slide 25 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 2 Solve equations containing an unknown factor.

26. Suppose we have to design a room in a house, and we know we want the area to be 150 square feet and the length to be 15 feet. What must the width be? Slide 26 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley We know the area of a rectangle is found by the formula A = l w, so we can write an equation with the width, w as an unknown factor.

27. Slide 27 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley To solve for an unknown factor, write a related division equation in which the product is divided by the known factor. Procedure

28. Slide 28 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 7 Solve and check. a. x 16 = 208

29. Slide 29 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 7 Solve and check. b. 14 n = 0

30. Slide 30 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 7 Solve and check. c. y 0 = 5

31. Slide 31 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 3 Solve applications involving division.

32. Slide 32 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 8 An egg farmer has 4394 eggs to distribute into packages of a dozen each. How many packages can be made? How many eggs will be left?

33. Slide 33 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 4 Find the square root of a perfect square.

34. Suppose we want to design a fenced area for our pet dog. The veterinarian tells us that the dog needs 400 square feet of area to stay healthy. If we make the fenced area a square, how long must each side be? Slide 34 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 400 ft. 2 ? ?

35. A square is a special rectangle in which the length and width are the same. Recall that to calculate the area of a rectangle, we multiply the length by the width. In the case of a square, since the length and width are the same, we multiply a number by itself, or square the number to get the area. Note the two meanings of the word square. Slide 35 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 400 ft. 2 ? ?

36. Slide 36 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Definition Square: a. Geometric: A rectangle with all sides of equal length. b. Algebraic: To multiply a number by itself. So in designing the fence, to find the length of each side, we must find the number that can be squared to equal 400. We say this unknown number is a square root of 400.

37. Slide 37 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Definition Square root: A square root of a given number is a number whose square is the given number. Since 20 2 = 400, the square root of 400 is 20. Our fence should be 20 feet by 20 feet. The symbol for square root is the radical sign. The number we wish to find the square root of is called the radicand. Radical sign

38. Slide 38 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Definition Perfect square: A number that has a whole number square root. RootPerfect squareRootPerfect square 00 2 = 01010 2 = 100 11 2 = 11111 2 = 121 22 2 = 41212 2 = 144 33 2 = 91313 2 = 169 44 2 = 161414 2 = 196 55 2 = 251515 2 = 225 66 2 = 361616 2 = 256 77 2 = 491717 2 = 289 88 2 = 641818 2 = 324 99 2 = 811919 2 = 361

39. Slide 39 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley To find a square root of a given number, find a number whose square is the given number. Procedure

40. Slide 40 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 9 Find the square root 169, 144, 81.

41. Slide 41 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 5 Solve applications involving square roots.

42. Slide 42 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 10 In building your new home, you decide to include an office. The plans can be developed with an office that has an area between 200 square feet and 250 square feet. What must the length and width of the office be if the area is to be a perfect square. Solution: