Chapter 4—Divisibility, Factors, and Prime Factorization (4-1) (4-2)
WORD BANK:
divisible
factor
product
multiple
prime
composite
One integer is______by another if there is no remainder when you divide. A number is a ______of another if it divides that number with no remainder. A ______is the result obtained by multiplying two numbers or variables. A ______is the product of a given number and an integer. A ______number is divisible by only itself and 1. A number is ______if it has more than itself and 1 as factors.
An integer is divisible by:
2 if it ends with 0, 2, 4, 6, 8
5 if it ends with a 0, 5
10 if it ends with a 0
3 if the sum of its digits is divisible by 3
6 if it’s divisible by 2 and 3
9 if the sum of its digits is divisible by 9
Practice: (pp 180+181) Is each number is divisible by 2, 3, 4, 5, 6, 8, 9, 10, or none? Prove it.
#23) 131 #24) 288
#26) 52 #28) 4,805
8,148 660,000
List all of the factor pairs for each:
#32) 28 #38) 72
Answers: 23) 131 is a prime #. It is only divisible by itself and 1; 24) 288 is divisible by 2, 3, 4, 6, and 9; 26) 52 is divisible by 2; 28) 4,805 is divisible by 5; 8,248 is divisible by 2, 3, 4, and 6; 660,000 is divisible by 2, 3, 4, 5, 6, 8, and 10; 32) 28’s factor pairs are (2, 16), and (4, 7); 38) 72’s factor pairs are (2, 36), (3, 14), (4, 18), (6, 12), and (8, 9)
Chapter 4—Exponents + the Greatest Common Factor or GCF (4-3)
WORD BANK:
exponent
base
power
prime factorization
prime factors
greatest common factor
GCF
prime
composite
You can use______to show repeated multiplication with the same integer. Example: “102” means 10 * 10 or 100. “10” is the______. “2” is the ______. Together they are read “10 to the second ______.” If you break down the factors repeatedly until you can’t break them down any more, than you will have the ______of the number. Write the ______in increasing order and use exponents to indicate repeated factors.
Example: 825= 11 * 3 * 5 * 5, so write: 3 * 52 * 11= 825
Order of Operations: PEMDAS
Please Excuse My Dear Aunt Sally Parenthesis ( ) Exponents ^2 Multiplication * Division /
Addition +
Subtraction -
Practice: (pp184+185) Simplify each expression:
#30) 15 + (4+6)2 ÷ 5 #31) (- 4)(- 6)2 (2)
#33) (12- 3)2 ÷ (22 – 14)
Evaluate each expression:
#35) 5k2, for k = 1.2 #36) 8 – x3, for x = - 2
#38) 4(2y – 3) 2, for y = 5
Answers: 30) 35; 31) – 288; 33) 27; 35) 7.2; 36) 0; 38) 98
Chapter 4—Exponents + the Greatest Common Factor or GCF (4-3)
A ______number is an integer with only two factors: itself and one.
A ______number is an integer that has more than two factors. Two composite numbers may have the same factors. The largest of these is called the ______(GCF). You can use the GCF to simplify fractions. You can use a factor tree to find prime factorizations.
Example: Prime Factorization of 24:
Factors of 24 Factors of 12
Venn Diagrams can be used to sort factors, showing which are common to both integers.
Practice: (p.189) Write the prime factorization of each number:
#12) 42 #13) 360
What are the common factors of both? ______
What is the greatest common factor?______
Answers: 12) 42= 2 * 3* 7; 13) 360 = 23 * 32 * 5. The common factors are 2 and 3. The GCF is 2 * 3 or 6.
Chapter 4—Simplifying Fractions (4-1) (4-2)
WORD BANK:
equivalent
simplify
______fractions describe the same part of a whole. You can find equivalent fractions by multiplying or dividing the denominator and numerator by the same number or variable. A fraction is in simplest form when the numerator and denominator have no common factors. You can use the GCF to write a fraction in simplest form. To ______a fraction you must divide the numerator and denominator by the same number.
Practice: (pp 194+195) Simplify each fraction:
#17) 24x/16 #18) 8 pr/12p
#19) 14a2/24a #21) 40ab2/5ab
#34) 5c2d/ 15c #39) 6m2n2 /9mn2
Find two fractions equivalent to each fraction:
#23) 4/10 #24) 5/20
#26) 18/20 #27) 25/100
Add:
6 14/21 + 5 3/12 8/16 + 2 ¾
Subtract:
6 14/21 - 5 3/12 8/16 - 2 ¾
Answers: 17) 3x/2; 18) 2r/3; 19) 7a/12; 21) 8b; 34) cd/3; 39) 2m/3; 23) 2/5 and 8/20; 24) ¼ and 10/40; 26) 9/10 and 36/40; 27) ¼ and 50/200; Add: 11 11/12; 3 ¼; Subtract: 1 5/12; - 2 ¼
Chapter 4—Accounting for All Possibilities: Combinations and Permutations (4-5)
There are four main ways of systematically finding all possibilities:
1. Make an organized or systematic list going in order through each of the possibilities.
2. Make a tree diagram with branches for each possibility.
3. Make a diagram (or net) with edges, or lines, showing each possible combination; then count the edges.
4. Make a table.
Practice: (pp. 199+200) Find all of the possibilities by making a list, using a tree diagram, drawing a net, and making a table:
#8) Using the paths shown, how many different ways can Jill walk to Trisha’s house (going only north or east)? How do you know that you have them all?
Trisha’s house
Jill’s house
N or E→
List:
Tree Diagram:
Net:
Table:
1st / next / next / lastAnswer: There are 6 ways: EENN, NNEE, ENEN, ENNE, NENE, NEEN.
Chapter 4—Rational Numbers (4-6)
WORD BANK:
rational number
irrational number
A______number is any number you can write as a quotient of two integers a/b, where b is not zero. All integers (whole numbers whether positive or negative) are rational numbers.
An irrational number is any number that cannot be expressed as a finite or repeating decimal. In other words, any number that keeps on going and going and going without repeating.
When evaluating a fraction containing a variable, remember that a fraction bar is a grouping symbol, so you must simplify the numerator and denominator separately first; then simplify the fraction as a whole.
Practice: (pp. 203+204) Graph each rational number on the number line:
1/10, - 3/5, - 5/9, - 4/4, - 2/3, 4/7, 1/5, 2.5, 50%
______
Evaluate each for a = - 4, b = - 6, then simplify.
#18) 2a + b/20 #20) b – a/3b
#33) m/n, for m = - 2, and n = 8 m/n, for m = 2, and n = - 8
Answers: - 4/4, - 2/3, - 3/5, - 5/9, 1/10, 1/5, 50%, 4/7, 2.5; 18) - 7/10; 20) +1/9; 33) – 4 and -4
Chapter 4—Exponents in Multiplication and Division (4-7) (4-8)
In general , to multiply numbers with the same base, add the exponents.
Example: 23 * 24 = 23+4 = 27
Practice: (p 207)
#4) 102 • 105 #5) 22 • 25
#8) (3)2 • (2)3 • 2 • 3 #9) x • y • y • x5 • y3
#10) 7b3 • 4b4 #16) – 7x6 • - 5x8
#17) – 5d5 • 6d 2
Answers: 4) 107 or 10,000,000; 5) 27 or 128; 8) 432 or 24• 33; 9) x6• y5; 10) 28b7; 16) 35x14; -30d7
Chapter 4—Exponents in Multiplication and Division (4-7) (4-8)
To find a power of a power for numbers with the same base, multiply the exponents.
Example: (23)4 = 23*4 = 212
Practice: (p 207) Simplify each expression:
#19) (103)2 #25) (x5)7
(12)5
Practice: (pp 207+208) Complete each inequality or equation using <, > or = :
#34) 252 ______(52)2 #36) (43 • 42)3 ____ 49
#39) Marcos thinks that x4 + x4 simplifies to 2x4. Doug thinks that x4 + x4 simplifies to x8. Which result is correct? Explain.
What would x4 • x4 be? How do you know?
Answers: 19) 105 or 100,000; 25) x12; 34) 252 = (52)2; 36) Marcos is correct. X4 + x4 = 2x4; x4 • x4 = x 8 because x4 means x • x • x • x, so if you did this twice, you’d have x • x • x • x • x • x • x • x or x8
Chapter 4—Exponents in Multiplication and Division (4-7) (4-8)
To divide numbers with the same base, subtract the exponents.
Example: 45 = 45-2 = 43
42
Any number raised to the zero power is equal to 1.
Example: 30 = 1 20 = 1 100 = 1
Have a negative exponent? Change your altitude!
Example: 3 - 2 = 1 OR 1 10 - 2 = 1 OR 1__
32 9 102 100
Practice: (pp 213+214) Simplify each expression:
#1) 25/22 #7) x7/x3
#9) 18x20/18x20 #21) b5/b8
Write each as an expression without the fraction bar:
#25) y4/y7 #26) a2 b4/a8 b2
#28) xy2/x4y9 #33) y ( ) = y - 4
y9
Answers: 1) 23; 7) x4; 9) 1; 21) 1/b3 or b -3; 25) 1/y3 or y – 3; 26) b2/a6 or b2a – 6; 28) 1/x3 y7 or x – 3 y - 7; 33) 1/y4 or y - 4
Chapter 4—Scientific Notation (4-9)
Scientific Notation / Expanded Notation / Standard Notation5 X 102 / 5 X 100 / 500
5 X 101 / 5 X 10 / 50
5 X 100 / 5 X 1 / 5
5 X 10 – 1 / 5 X 0.1 / 0.5
5 X 10 – 2 / 5 X 0.01 / 0.05
______provides a way to write very large and very small numbers using powers of ten. You write the number in scientific notation as a product of two factors, one of which is 10. In scientific notation, you can use a negative exponent to represent a number between 0 and 1.
Example: 7,500,000,000,000 = 7.5 * 1012
.0000000000075 = 7.5 * 10 - 12
The digit in front of the decimal must be greater than 1 but less than 10. The exponent is the number of digits that come after (or in the case of negative exponents: before) the decimal point in the original number. In the example above the exponent is 12 because there are twelve digits after the 7.
You can change expressions from scientific notation to ______by simplifying the product of the two factors. In the example above, 7,500,000,000,000 is the number in standard notation.
To multiply with scientific notation, add the exponents. To divide with scientific notation, subtract the exponents.
Practice: (pp 219+220) Write in scientific notation:
#2) 555,900, 000 #3) .0000631
Write in standard notation:
#12) 2.75 X 108 #13) 6.0502 X 10 - 3
Answers: 2) 5.559 x 108; 3) 6.31 x 10 - 5; 12) 275,000,000; 13) .0060502
Chapter 4—Scientific Notation (4-9)
Practice: (pp. 219+220) Order from least to greatest:
#19) 253 X 10 – 9, 3.7 X 10 – 8, 12.9 X 10-7
#32) 55.8 X 10 – 5, 782 x 10 – 8, 9.1 X 10 - 5, 1,009 X 102, 0.8 X 10 - 4
Multiply then write in scientific notation:
#21) (5 X 106)(6 X 102) #22) (4.3 X 103)(2 X 10 - 8)
#23) (9 X 10 - 3)(7 X 108)
Answers: 19) .000000037 or 3.7 x 10 – 8, then .000000253 or 253 x 10 - 9, then .00000129 or 12.9 x 10 – 7;
32) 782 x 10 – 8 or .00000782, 9.1 x 10 – 5 or .000091, 55.8 x 10 – 5 or .000558, 0.8 x 10 – 4 or .0008, 1,009 x 102 or 10,090; 21) 3.0 x 109 or 300,000,000; 22) 8.6 x 10 – 5 or .000086; 23) 6.3 x 106 or 6,300,000