1. Pre-Algebra Divisibility and Factors Lesson 4-1 Objectives: 1. to use divisibility tests 2. to find factors

2. Pre-Algebra Divisibility and Factors Lesson 4-1 Tips:the product of two integers is an integer, and both integers are factors of the product. Moreover, both integers divide the product, and the product is said to be divisible by each integer.

3. Pre-Algebra Is the first number divisible by the second? Divisibility and Factors Lesson 4-1 a. 1,028 by 2 Yes; 1,028 ends in 8. b. 572 by 5 No; 572 doesn’t end in 0 or 5. c. 275 by 10 No; 275 doesn’t end in 0.

4. Pre-Algebra Is the first number divisible by the second? Divisibility and Factors Lesson 4-1 a. 1,028 by 3 No; 1 + 0 + 2 + 8 = 11; 11 is not divisible by 3. b. 522 by 9 Yes; 5 + 2 + 2 = 9; 9 is divisible by 9.

5. Pre-Algebra Ms. Washington’s class is having a class photo taken. Each row must have the same number of students. There are 35 students in the class. How can Ms. Washington arrange the students in rows if there must be at least 5 students, but no more than 10 students, in each row? Divisibility and Factors Lesson 4-1 1 35, 5 7 Find pairs of factors of 35. There can be 5 rows of 7 students, or 7 rows of 5 students.

6. Pre-Algebra Exponents Lesson 4-2 Objectives: 1. to use exponents 2. to use the order of operations with exponents

7. Pre-Algebra Exponents Lesson 4-2 Tips:the exponent is placed to the upper right of the base, and it only applies to that base. If an exponent has as its base an expression, that expression must be written in parentheses.

8. Pre-Algebra Write using exponents. Exponents Lesson 4-2 b. –5 x x y y x a. (–11)(–11)(–11)(–11) –5 x x x y y Rewrite the expression using the Commutative and Associative Properties. –5x 3 y 2 Write x x x and y y using exponents. (–11) 4 Include the negative sign within parentheses.

9. Pre-Algebra Suppose a certain star is 10 4 light-years from Earth. How many light-years is that? Exponents Lesson 4-2 10 4 = 10 10 10 10 The exponent indicates that the base 10 is used as a factor 4 times. = 10,000 light-yearsMultiply. The star is 10,000 light-years from Earth.

10. Pre-Algebra Exponents Lesson 4-2 a. Simplify 3(1 + 4) 3. b. Evaluate 7(w + 3) 3 + z, for w = –5 and z = 6. 3(1 + 4) 3 = 3(5) 3 Work within parentheses first. = 3 125Simplify 5 3. = 375Multiply. = 7(–2) 3 + 6Work within parentheses. Replace w with –5 and z with 6.7(w + 3) 3 + z = 7(–5 + 3) 3 + 6 = 7(–8) + 6Simplify (–2) 3. = –56 + 6Multiply from left to right. = –50Add.

11. Pre-Algebra Prime Factorization and Greatest Common Factor Lesson 4-3 Objectives: 1. to find the prime factorization of a number 2. to find the greatest common factor (GCF) of two or more numbers Tips:remember a factor is a number that divides evenly into another number with a remainder of zero.

12. Pre-Algebra Prime Factorization and Greatest Common Factor Lesson 4-3 State whether each number is prime or composite. Explain. a. 46 Composite; 46 has more than two factors, 1, 2, 23, and 46. b. 13 Prime; 13 has exactly 2 factors, 1 and 13.

13. Pre-Algebra Prime Factorization and Greatest Common Factor Lesson 4-3 Use a factor tree to write the prime factorization of 273. 273 273 = 3 7 13 PrimeStart with a prime factor. Continue branching. 3 91 7 13PrimeStop when all factors are prime. 3 7 13 Write the prime factorization.

14. Pre-Algebra Prime Factorization and Greatest Common Factor Lesson 4-3 Find the GCF of each pair of numbers or expressions. a. 24 and 30 The GCF of 24 and 30 is 6. b. 36ab 2 and 81b The GCF of 36ab 2 and 81b is 9b. 24 = 2 3 3Write the prime factorizations. 30 = 2 3 5 GCF = 2 3 = 6 Use the lesser power of the common factors. GCF = 3 2 b = 9b Use the lesser power of the common factors. 36ab 2 = 2 2 3 2 a b 2 Write the prime factorizations. 81b =3 4 b Find the common factors.

15. Pre-Algebra Simplifying Fractions Lesson 4-4 Objectives: 1. to find equivalent fractions 2. to write fractions in simplest for Tips:always check your answers and the steps involved in finding the answers

16. Pre-Algebra Simplifying Fractions Lesson 4-4 Find two fractions equivalent to. 18 21 a. 18 21 b. 18 21 18 2 21 2 = 18 ÷ 3 21 ÷ 3 = = 36 42 6767 = The fractions and are both equivalent to. 6767 36 42 18 21

17. Pre-Algebra Simplifying Fractions Lesson 4-4 You learn that 21 out of the 28 students in a class, or, buy their lunches in the cafeteria. Write this fraction in simplest form. 21 28 The GCF of 21 and 28 is 7. 21 28 21 ÷ 7 28 ÷ 7 =Divide the numerator and denominator by the GCF, 7. 3434 =Simplify. 3434 of the students in the class buy their lunches in the cafeteria.

18. Pre-Algebra Simplifying Fractions Lesson 4-4 Write in simplest form. a. p2pp2p p2pp2p = p12p1p12p1 Divide the numerator and denominator by the common factor, p. 1212 = Simplify.

19. Pre-Algebra Simplifying Fractions Lesson 4-4 (continued) b. 14q 2 rs 3 8qrs 2 = Write as a product of prime factors. 14q 2 rs 3 8qrs 2 2 7 q q r s s s 2 2 2 q r s s 2 1 7 q 1 q r 1 s 1 s 1 s 2 1 2 2 q 1 r 1 s 1 s 1 Divide the numerator and denominator by the common factors. = Simplify. 7 q s 2 = Simplify. 7 q s 4 = 7qs 4 =

20. Pre-Algebra Rational Numbers Lesson 4-6 Objectives: 1. to identify and graph rational numbers 2. to evaluate fractions containing variables Tips:the quotient of two integers with the same sign is positive

21. Pre-Algebra Rational Numbers Lesson 4-6 Write two lists of fractions equivalent to. 2323 2323 4646 6969 = = = … Numerators and denominators are positive.= = = … Numerators and denominators are negative. 2323 –2 –3 –4 –6

22. Pre-Algebra Rational Numbers Lesson 4-6 Graph each rational number on a number line. a. b. 0.5 c. 0 d. 3434 – 1313 Additional Examples

23. Pre-Algebra Rational Numbers Lesson 4-6 A fast sports car can accelerate from a stop to 90 ft/s in 5 seconds. What is its acceleration in feet per second per second (ft/s 2 )? Use the formula a =, where a is acceleration, f is final speed, i is initial speed, and t is time. f – i t a = f – i t Use the acceleration formula. The car’s acceleration is 18 ft/s 2. 90 – 0 5 Substitute.= 90 5 Subtract.= 18=Write in simplest form.

24. Pre-Algebra Exponents and Multiplication Lesson 4-7 Objectives: 1. to find areas of rectangles 2. to find areas of parallelograms

25. Pre-Algebra Exponents and Multiplication Lesson 4-7 Tips:look for two segments that form a right angle when determining a base and height

26. Pre-Algebra Exponents and Multiplication Lesson 4-7 Simplify each expression. a. 5 2 5 3 = 5 5 b. x 5 x 7 y 2 y 5 2 5 3 = 5 2 + 3 Add the exponents of powers with the same base. = 3,125Simplify. x 5 x 7 y 2 y = x 5 + 7 y 2 + 1 Add the exponents of powers with the same base. = x 12 y 3 Simplify.

27. Pre-Algebra Exponents and Multiplication Lesson 4-7 Simplify 3a 3 (–5a 4 ). 3a 3 (–5a 4 ) = 3 (–5) a 3 a 4 Use the Commutative Property of Multiplication. Add the exponents.= –15a 3 + 4 Simplify.= –15a 7

28. Pre-Algebra Exponents and Multiplication Lesson 4-7 Simplify each expression. a. (2 3 ) 3 = (2) 9 Simplify the exponent. (2 3 ) 3 = (2) 3 3 Multiply the exponents. = 512Simplify. b. (g 5 ) 4 (g 5 ) 4 = g 5 4 Multiply the exponents. = g 20 Simplify the exponent.

29. Pre-Algebra Exponents and Division Lesson 4-8 Objectives: 1. to find areas of rectangles 2. to find areas of parallelograms

30. Pre-Algebra Exponents and Division Lesson 4-8 Tips:look for two segments that form a right angle when determining a base and height

31. Pre-Algebra Exponents and Division Lesson 4-8 Simplify each expression. a.a. b.b. w 18 w 13 = 4 4 Simplify the exponent. = 256Simplify. = w 5 Simplify the exponent. 4 12 4 8 4 12 4 8 = 4 12 – 8 Subtract the exponents. = w 18 – 13 w 18 w 13 Subtract the exponents.

32. Pre-Algebra Exponents and Division Lesson 4-8 Simplify each expression. a.a. (–12) 73 = 1 = (–12) 0 Simplify. b.b. 8s 20 32s 20 (–12) 73 = (–12) 73 – 73 Subtract the exponents. 1414 = 1Simplify s 0. 1414 =Multiply. 8 32 8s 20 32s 20 1414 =Subtract the exponents. Simplify.s0s0

33. Pre-Algebra Exponents and Division Lesson 4-8 Simplify each expression. a.a. 6 12 6 14 b.b. z 4 z 15 = 6 –2 6 12 6 14 = 6 12 – 14 Subtract the exponents. = 162162 Write with a positive exponent. 1 36 =Simplify. = z –11 z 4 z 15 = z 4 – 15 Subtract the exponents. = 1 z 11 Write with a positive exponent.

34. Pre-Algebra Exponents and Division Lesson 4-8 Write without a fraction bar. a 2 b 3 ab 15 a 2 b 3 ab 15 = a 2 – 1 b 3 – 15 Use the rule for Dividing Powers with the Same Base. = ab –12 Subtract the exponents.

35. Pre-Algebra Scientific Notation Lesson 4-9 Objectives: 1. to find areas of rectangles 2. to find areas of parallelograms Tips:look for two segments that form a right angle when determining a base and height

36. Pre-Algebra Scientific Notation Lesson 4-9 About 6,300,000 people visited the Eiffel Tower in the year 2000. Write this number in scientific notation. 6,300,000 6.3Drop the zeros after the 3. 6.3  10 6 You moved the decimal point 6 places. The original number is greater than 10. Use 6 as the exponent of 10. Move the decimal point to get a decimal greater than 1 but less than 10. 6 places

37. Pre-Algebra Scientific Notation Lesson 4-9 Write 0.00037 in scientific notation. 0.00037Move the decimal point to get a decimal greater than 1 but less than 10. 4 places 3.7  10 –4 You moved the decimal point 4 places. The original number is less than 1. Use –4 as the exponent of 10. 3.7Drop the zeros before the 3.

38. Pre-Algebra Scientific Notation Lesson 4-9 Write each number in standard notation. a. 3.6  10 4 3.6000Write zeros while moving the decimal point. 36,000Rewrite in standard notation. b. 7.2  10 –3 007.2 0.0072Rewrite in standard notation. Write zeros while moving the decimal point.

39. Pre-Algebra Scientific Notation Lesson 4-9 Write each number in scientific notation. a. 0.107  10 12 0.107  10 12 = 1.07  10 –1  10 12 Write 0.107 as 1.07  10 –1. = 1.07  10 11 Add the exponents. b. 515.2  10 –4 515.2  10 –4 = 5.152  10 2  10 –4 Write 515.2 as 5.152  10 2. = 5.152  10 –2 Add the exponents.

40. Pre-Algebra Scientific Notation Lesson 4-9 Write each number in scientific notation. 0.035  10 4 0.69  10 2 3.5  10 2 710  10 –1 7.1  106.9  10 Order the powers of 10. Arrange the decimals with the same power of 10 in order. 6.9  107.1  103.5  10 2 Write the original numbers in order. 0.69  10 2, 710  10 –1, 0.035  10 4 Order 0.035  10 4, 710  10 –1, and 0.69  10 2 from least to greatest.

41. Pre-Algebra Scientific Notation Lesson 4-9 Multiply 4  10 –6 and 7  10 9. Express the result in scientific notation. Use the Commutative Property of Multiplication. (4  10 –6 )(7  10 9 ) = 4  7  10 –6  10 9 = 28  10 –6  10 9 Multiply 4 and 7. = 28  10 3 Add the exponents. = 2.8  10 1  10 3 Write 28 as 2.8  10 1. = 2.8  10 4 Add the exponents.

42. Pre-Algebra Scientific Notation Lesson 4-9 In chemistry, one mole of any element contains approximately 6.02  10 23 atoms. If each hydrogen atom weighs approximately 1.67  10 –27 kg, approximately how much does one mole of hydrogen atoms weigh? (6.02  10 23 )(1.67  10 –27 ) Multiply number of atoms by weight of each. = 6.02  1.67  10 23  10 –27 Use the Commutative Property of Multiplication. = 10.1  10 –4 Add the exponents. = 1.01  10 1  10 –4 Write 10.1 as 1.01  10 1. = 1.01  10 –3 Add the exponents.Multiply 6.02 and 1.67. 10.1  10 23  10 –27 One mole of hydrogen atoms weighs approximately 1.01  10 –3 kg.