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

2. Divisibility and Factors
Lesson 4-1
New Terms: divisible – one integer is divisible by another if the remainder is 0 when you divide
2. factor – one integer is a factor of another integer (not zero) if it divides that integer with a remainder zero.
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. Divisibility and Factors
Lesson 4-1
Is the first number divisible by the second?
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. Divisibility and Factors
Lesson 4-1
Is the first number divisible by the second?
a. 1,028 by 3
No; = 11; 11 is not divisible by 3.
b. 522 by 9
Yes; = 9; 9 is divisible by 9.

5. Divisibility and Factors
Lesson 4-1
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?
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. Exponents Lesson 4-2 Objectives: 1. to use exponents
2. to use the order of operations with exponents

7. Exponents
Lesson 4-2
New Terms: 1. exponents – you can use exponents to show repeated multiplication
2. power – a power has two parts, a base and an exponent
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. Include the negative sign within parentheses.
Exponents
Lesson 4-2
Write using exponents.
a. (–11)(–11)(–11)(–11)
(–11)4
Include the negative sign within parentheses.
b. –5 • x • x • y • y • x
–5 • x • x • x • y • y
Rewrite the expression using the Commutative and Associative Properties.
–5x3y2
Write x • x • x and y • y using exponents.

9. The exponent indicates that the base 10 is used as a factor 4 times.
Exponents
Lesson 4-2
Suppose a certain star is 104 light-years from Earth. How many light-years is that?
104 = 10 • 10 • 10 • 10
The exponent indicates that the base 10 is used as a factor 4 times.
= 10,000 light-years
Multiply.
The star is 10,000 light-years from Earth.

10. Work within parentheses first.
Exponents
Lesson 4-2
a. Simplify 3(1 + 4)3.
3(1 + 4)3 = 3(5)3
Work within parentheses first.
= 3 • 125
Simplify 53.
= 375
Multiply.
b. Evaluate 7(w + 3)3 + z, for w = –5 and z = 6.
Replace w with –5 and z with 6.
7(w + 3)3 + z = 7(–5 + 3)3 + 6
= 7(–2)3 + 6
Work within parentheses.
= 7(–8) + 6
Simplify (–2)3.
= –56 + 6
Multiply from left to right.
= –50
Add.

11. 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
New Terms: prime number – is an integer greater than 1 with exactly two positive factors, 1 and itself
2. composite number – is an integer greater than 1 with more than two positive factors
3. greatest common factor – factors that are the same for two or more numbers or expressions are common factors. The greatest of these is called the GCF.
Tips: remember a factor is a number that divides evenly into another number with a remainder of zero.

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

14. Prime Factorization and Greatest Common Factor
Lesson 4-3
Find the GCF of each pair of numbers or expressions.
a. 24 and 30
24 = 23 • 3
Write the prime factorizations.
30 = 2 • 3 • 5
Find the common factors.
GCF = 2 • 3
= 6
Use the lesser power of the common factors.
The GCF of 24 and 30 is 6.
b. 36ab2 and 81b
36ab2 = 22 • 32 • a • b2
Write the prime factorizations.
81b =
34 • b
Find the common factors.
GCF = 32 • b
= 9b
Use the lesser power of the
common factors.
The GCF of 36ab2 and 81b is 9b.

15. Simplifying Fractions
Lesson 4-4
Objectives: 1. to find equivalent fractions
2. to write fractions in simplest for
New Terms: simplest form – a fraction is in simplest form when the numerator and the denominator have no common factors other than 1.
Tips: always check your answers and the steps involved in finding the answers

16. Simplifying Fractions
Lesson 4-4 18 21
Find two fractions equivalent to 18 21
18 • 2
21 • 2 =
a.  = 36 42 18 21
18 ÷ 3
21 ÷ 3 =
b.  6 7 =
The fractions and are both equivalent to 6 7 36 42 18 21

17. 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. 3 4 =
Simplify. 3 4
of the students in the class buy their lunches in the cafeteria.

18. Simplifying Fractions
Lesson 4-4
Write in simplest form. p 2p a. p 2p = p1
2p1
Divide the numerator and denominator by the common factor, p. 1 2 =
Simplify.

19. Simplifying Fractions
Lesson 4-4
(continued) b.
14q2rs3
8qrs2 =
Write as a product of prime factors.
14q2rs3
8qrs2
2 • 7 • q • q • r • s • s • s
2 • 2 • 2 • q • r • s • s
21 • 7 • q1 • q • r1 • s1 • s1 • s
21 • 2 • 2 • q1 • r1 • s1 • s1
Divide the numerator and denominator by the common factors. =
Simplify.
7 • q • s
2 • 2 =
Simplify.
7 • q • s 4 =
7qs 4 =

20. Problem Solving Strategy: Solve a Simpler Problem
Lesson 4-5
Additional Examples
Aaron, Chris, Maria, Sonia, and Ling are on a class committee. They want to choose two members to present their conclusions to the class. How many different groups of two members can they form?

21. Problem Solving Strategy: Solve a Simpler Problem
Lesson 4-5
Additional Examples
(continued)
First, pair Aaron with each of the four other committee members.
Aaron
Maria
Chris
Sonia
Ling
Next, pair Chris with each of the three members left.
Chris
Sonia
Maria
Ling
Sonia
Ling
Maria
Since Aaron and Chris have already been paired, you don’t need to count them again. Repeat for the rest of the committee members.
Each successive tree has one less branch.
There are 10 different groups of two committee members.

22. Rational Numbers Lesson 4-6
Objectives: 1. to identify and graph rational numbers
2. to evaluate fractions containing variables
New Terms: rational number – is any number you can write as a fraction
Tips: the quotient of two integers with the same sign is positive

23. Write two lists of fractions equivalent to .
Rational Numbers
Lesson 4-6
Write two lists of fractions equivalent to . 2 3 2 3 4 6 9
= = = … Numerators and denominators are positive.
= = = … Numerators and denominators are negative. 2 3 –2 –3 –4 –6

24. Graph each rational number on a number line.
Rational Numbers
Lesson 4-6
Additional Examples
Graph each rational number on a number line. 3 4 a. –
b. 0.5
c. 0 1 3 d.

25. Use the acceleration formula.
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/s2)? 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.
90 – 0 5
Substitute. = 90 5
Subtract. = 18 =
Write in simplest form.
The car’s acceleration is 18 ft/s2.

26. Exponents and Multiplication
Lesson 4-7
Objectives: 1. to multiply powers with the same base
2. to find a power of a power

27. Exponents and Multiplication
Lesson 4-7
Tips: when in doubt, write it out.

28. Exponents and Multiplication
Lesson 4-7
Simplify each expression.
a. 52 • 53
52 • 53 =
Add the exponents of powers with the
same base.
= 55
= 3,125
Simplify.
b. x5 • x7 • y2 • y
x5 • x7 • y2 • y = x5 + 7 • y2 + 1
Add the exponents of powers with the same base.
= x12y3
Simplify.

29. Exponents and Multiplication
Lesson 4-7
Simplify 3a3 • (–5a4).
3a3 • (–5a4) = 3 • (–5) • a3 • a4
Use the Commutative Property
of Multiplication.
Add the exponents.
= –15a3 + 4
Simplify.
= –15a7

30. Exponents and Multiplication
Lesson 4-7
Simplify each expression.
a. (23)3
(23)3 = (2)3 • 3
Multiply the exponents.
= (2)9
Simplify the exponent.
= 512
Simplify.
b. (g5)4
(g5)4 = g5 • 4 
Multiply the exponents.
= g20
Simplify the exponent.

31. Exponents and Division
Lesson 4-8
Objectives:
1. To divide expressions containing exponents
2. To simplify expressions with integer exponents

32. Exponents and Division
Lesson 4-8
Tips: Sometimes students think an expression with a negative exponent makes the expression negative, that is not true. A negative exponent makes the number smaller (unless the base is less than one but greater than zero).

33. Exponents and Division
Lesson 4-8
Simplify each expression.
412 48 a.
412 48
= 412 – 8
Subtract the exponents.
= 44
Simplify the exponent.
= 256
Simplify.
w18
w13 b.
= w18 – 13
w18
w13
Subtract the exponents.
= w5
Simplify the exponent.

34. Exponents and Division
Lesson 4-8
Simplify each expression.
(–12)73 a.
(–12)73
= (–12)73 – 73
Subtract the exponents.
= (–12)0
Simplify.
= 1 b.
8s20
32s20 8 32
8s20
32s20 1 4 =
Subtract the exponents. Simplify s0 1 4 =
• 1
Simplify s0. 1 4 =
Multiply.

35. Exponents and Division
Lesson 4-8
Simplify each expression.
612
614 a.
= 6–2
612
614
= 612 – 14
Subtract the exponents. = 1 62
Write with a positive exponent. 1 36 =
Simplify. z4
z15 b.
= z–11 z4
z15
= z4 – 15
Subtract the exponents. = 1
z11
Write with a positive exponent.

36. Exponents and Division
Lesson 4-8
Write without a fraction bar.
a2b3
ab15
a2b3
ab15
= a2 – 1b3 – 15  
Use the rule for Dividing Powers with the
Same Base.
= ab–12
Subtract the exponents.

37. Scientific Notation Lesson 4-9
Objectives: 1. to write and evaluate numbers in scientific notation
2. to calculate with scientific notation
New Terms: scientific notation – a way to write numbers using powers of 10
2. standard notation – write the number using all digit and decimal places
Tips: be careful with which way you move the decimal, think if the number is suppose to be smaller or larger

38. Drop the zeros after the 3.
Scientific Notation
Lesson 4-9
About 6,300,000 people visited the Eiffel Tower in the year Write this number in scientific notation.
6,300,000
Move the decimal point to get a decimal greater than 1 but less than 10.
6 places
6.3
Drop the zeros after the 3.
6.3 x 106
You moved the decimal point 6 places. The original number is
greater than 10. Use 6 as the exponent of 10.

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

40. Write each number in standard notation.
Scientific Notation
Lesson 4-9
Write each number in standard notation.
a. 3.6 x 104
3.6000
Write zeros while moving the decimal point.
36,000
Rewrite in standard notation.
b. 7.2 x 10–3
007.2
Write zeros while moving the decimal point.
0.0072
Rewrite in standard notation.

41. Write each number in scientific notation.
Lesson 4-9
Write each number in scientific notation.
a. 0.107 x 1012
0.107 x 1012 = 1.07 x 10–1 x 1012
Write as
1.07  10–1.
= 1.07 x 1011
Add the exponents.
b. 515.2 x 10–4
515.2 x 10–4 = x 102 x 10–4
Write as  102.
= x 10–2
Add the exponents.

42. Order 0.035 x 104, 710 x 10–1, and 0.69 x 102 from least to greatest.
Scientific Notation
Lesson 4-9
Order x 104, 710 x 10–1, and 0.69 x 102 from least to greatest.
Write each number in scientific notation.
0.035 x 104
0.69 x 102
3.5 x 102
710 x 10–1
7.1 x 10
6.9 x 10
Order the powers of 10. Arrange the decimals with the same power of 10 in order.
6.9 x 10
7.1 x 10
3.5 x 102
Write the original numbers in order.
0.69 x 102, 710 x 10–1, x 104

43. Use the Commutative Property of Multiplication.
Scientific Notation
Lesson 4-9
Multiply 4 x 10–6 and 7 x 109. Express the result in scientific notation.
Use the Commutative Property of Multiplication.
(4 x 10–6)(7 x 109) = 4 x 7 x 10–6 x 109  
= 28 x 10–6 x 109
Multiply 4 and 7.
= 28 x 103
Add the exponents.
= 2.8 x 101 x 103
Write 28 as 2.8  101.
= 2.8 x 104
Add the exponents.

44. Multiply number of atoms by weight of each.
Scientific Notation
Lesson 4-9
In chemistry, one mole of any element contains approximately 6.02 x 1023 atoms. If each hydrogen atom weighs approximately 1.67 x 10–27 kg, approximately how much does one mole of hydrogen atoms weigh?
(6.02 x 1023)(1.67 x 10–27)
Multiply number of atoms by weight of each.
= 6.02 x 1.67 x 1023 x 10–27
Use the Commutative Property of Multiplication.
Multiply 6.02 and 1.67.
10.1 x 1023 x 10–27
= 10.1 x 10–4
Add the exponents.
= 1.01 x 101 x 10–4
Write 10.1 as 1.01  101.
= 1.01 x 10–3
Add the exponents.
One mole of hydrogen atoms weighs approximately 1.01 x 10–3 kg.