CAHSEE on Target UC Davis, School and University Partnerships


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3 UC Davis, School and University Partnerships CAHSEE on Target Mathematics Curriculum Published by The University of California, Davis, School/University Partnerships Program 2006 Director Sarah R. Martinez, School/University Partnerships, UC Davis Developed and Written by Syma Solovitch, School/University Partnerships, UC Davis Editor Nadia Samii, UC Davis Nutrition Graduate Reviewers Faith Paul, School/University Partnerships, UC Davis Linda Whent, School/University Partnerships, UC Davis The CAHSEE on Target curriculum was made possible by funding and support from the California Academic Partnership Program, GEAR UP, and the University of California Office of the President. We also gratefully acknowledge the contributions of teachers and administrators at Sacramento High School and Woodland High School who piloted the CAHSEE on Target curriculum. Copyright The Regents of the University of California, Davis campus, All Rights Reserved. Pages intended to be reproduced for students activities may be duplicated for classroom use. All other text may not be reproduced in any form without the express written permission of the copyright holder. For further information, please visit the School/University Partnerships Web site at:
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5 Introduction to the CAHSEE The CAHSEE stands for the California High School Exit Exam. The mathematics section of the CAHSEE consists of 80 multiplechoice questions that cover 53 standards across 6 strands. These strands include the following: Number Sense (14 Questions) Statistics, Data Analysis & Probability (12 Questions) Algebra & Functions (17 Questions) Measurement & Geometry (17 Questions) Mathematical Reasoning (8 Questions) Algebra 1 (12 Questions) What is CAHSEE on Target? CAHSEE on Target is a tutoring course specifically designed for the California High School Exit Exam (CAHSEE). The goal of the program is to pinpoint each student s areas of weakness and to then address those weaknesses through classroom and small group instruction, concentrated review, computer tutorials and challenging games. Each student will receive a separate workbook for each strand and will use these workbooks during their tutoring sessions. These workbooks will present and explain each concept covered on the CAHSEE, and introduce new or alternative approaches to solving math problems. What is Number Sense? Number Sense is the understanding of numbers and their relationships. The Number Sense Strand concepts that are tested on the CAHSEE can be divided into five major topics: Integers & Fractions; Exponents; Word Problems; Percents; and Interest. These topics are presented as separate units in this workbook. 1
6 Unit 1: Integers & Fractions On the CAHSEE, you will be given several problems involving rational numbers (integers, fractions and decimals). Integers are whole numbers; they include... positive whole numbers {1, 2, 3,... } negative whole numbers { 1, 2, 3,... } and zero {0}. Positive and negative integers can be thought of as opposites of one another. A. Signs of Integers All numbers are signed (except zero). They are either positive or negative. When adding, subtracting, multiplying and dividing integers, we need to pay attention to the sign (+ or ) of each integer. Example: 53 = Example: = Example: = Whether it s written or not, every number has a sign: Example: 5 means +5 2
7 Signed Numbers in Everyday Life Signed numbers are used in everyday life to describe various situations. Often, they are used to indicate opposites: Altitude: The elevator went up 3 floors (+3) and then went down 5 floors (5). Weight: I lost 20 pounds (20) but gained 10 back (+10). Money: I earned $60 (+60) and spent $25 (25). Temperature: The temperature rose 5 degrees (+5) and then fell 2 degrees (2). Sea Level: Jericho, the oldest inhabited town in the world, lies 853 feet below sea level (853), making it the lowest town on earth. Mount Everest is the highest mountain in the world, standing at 8850 meters (+8850), nearly 5.5 miles above sea level. Can you think of any other examples of how signed numbers are used in life? 3
8 i. Adding Integers When adding two or more integers, it is very important to pay attention to the sign of each integer. Are we adding a positive or negative integer? We can demonstrate this concept with a number line. Look at the two examples below. In the first example, we add a positive 3 (+3) to 2. Example: = In this second example, we add a negative 3 (3) to 2. Example: 2 + ( 3) = As you can see, we get a very different answer in this second problem To add integers using a number line, begin with the first number in the equation. Place your finger on that number on the number line. Look at the value and sign of the second number: if positive, move to the right; if negative, move to the left. (If a number does not have a sign, this means it is positive.) With your finger, move the number of spaces indicated by the second number. Example: 1 + (2) = 4
9 On Your Own 2 + (3) = 6 + (3) = 3 + (6) = = 5
10 Rules for Adding Signed Numbers (without a Number Line) A. Same Signs Find the sum Keep the sign B. Different Signs Find the difference Keep the sign of the larger number (# with larger absolute value) On Your Own 8 + (7) = = (13) + (9) = (+13) + (+9) = 21 + (21) = (21) + 21 = = = Add 10 and 5: Add (10), (+4), and (16): 6
11 ii. Subtracting Integers We can turn any subtraction problem into an addition problem. Just change the subtraction sign () to an addition sign (+) and change the sign of the second number. Then solve as you would an addition problem. Example: 2  (+ 3) = Turn it from a subtraction problem to an addition problem; then change the sign of the second number: 2  (+ 3) = 2 + (3) Now solve as you would an addition problem. We can show this on a number line. Place your finger on that number on the number line. Look at the value and sign of the second number: if positive, move to the right; if negative, move to the left. With your finger, move the number of spaces indicated by the second number: Let's look at another problem: Answer: Example: 2  (3) = 2 + ( ) Answer: 7
12 On Your Own CAHSEE on Target 6 (+3) = 6 + ( ) = 3 (3) = 3 + ( ) = 5 (+1) = 5 + ( ) = 1 (+1) = 1 + ( ) = 8
13 Rules for Subtracting Signed Numbers (without a Number Line) Add its opposite! Draw the line and change the sign (of the second number), and follow the rules for addition. Example: 6 (4) Steps: Draw the line (to turn the minus sign into a plus sign): 6 + Change the sign of the second number: 6 + (+ 4) Now you have an addition problem. Follow the rules of adding numbers: =10 On Your Own: Draw the line and change the sign. Then solve the addition problem. 19 ( 13) = 17 (15) = 34 (9) = = (35) = 13  (+15) = = (+35) = Subtract (15) from (20): Subtract 4 from (14): 9
14 Signed Numbers Continued Look at the following problem: Example: = We can represent this problem on a number line: We begin at 1, move 3 spaces backwards (to the left) and then 5 spaces forwards (to the right). We arrive at + 3. When we are given a problem with three or more signed integers, we must work out, separately, the addition and subtraction for each integer pair: = = 2 Work out the addition or subtraction for the 1st 2 integers = Take the answer from above & add it to the last integer. On Your Own = = = 10
15 iii. Multiplying and Dividing with Signed Numbers Multiplying The product of two numbers with the same sign is positive. Example: 53 = 15 The product of two numbers with different signs is negative. Example: 5 3 = 15 Dividing The quotient of two numbers with the same sign is positive. Example: = 5 The quotient of two numbers with different signs is negative. Example: = 5 On Your Own (+8) ( 4) = ( 7) (7) = ( 8) (8) = (+7)(+8) = 36 (3) = 36 3 = 11
16 B. Absolute Value The absolute value of a number is its distance from 0. This distance is always expressed as a positive number, regardless if the number is positive or negative. It is easier to understand this by examining a number line: The absolute value of 5, expressed as 5, is 5 because it is 5 units from 0. We can see this on the number line above. The absolute value of 5, expressed as 5, is also 5 because it is 5 units from 0. Again, look at the number line and count the number of units from 0. On Your Own: Complete the chart. How far from zero is the number? Number Absolute Value x x
17 Finding the Absolute Value of an Expression On the CAHSEE, you may need to find the absolute value of an expression. To do this,... Evaluate the expression within the absolute value bars. Take the absolute value of that result. Perform any additional operations outside the absolute value bars. Example: = = = 10 On Your Own: Complete the chart = 55 = 5 5 = = = = = = = = = = = = = = = = = 13
18 Absolute Value Continued 2.5 While the absolute value of a number or expression will always be positive, the number between the absolute value bars can be positive or negative. Notice that in each case, the expression is equal to +8. You may be asked to identify these two possible values on the CAHSEE. Example: If x = 8, what is the value of x? For these types of problems, the answer consists of two values: the positive and negative value of the number. In the example above, the two values for x are 8 or 8. On Your Own 1. If y = 225, what is the value of y? or 2. If x = 1,233, what is the value of x? or 3. If m = 18, what is the value of m? or 4. If x = 12, what is the value of x? or 5. If y = 17, what is the value of y? or 14
19 C. Fractions A fraction means a part of a whole. Example: In the picture below, one of four equal parts is shaded: We can represent this as a fraction: 4 1 Fractions are expressed as one number over another number: 4 1 Every fraction consists of a numerator (the top number) and a denominator (the bottom number): A B Numerator Denominator Fractions mean division: B A = A B 1 = 1 4 = 4 1 = = 4 5 = 5 4 = = 1 2 = 2 1 =
20 i. Adding & Subtracting Fractions Same Denominator: Keep the denominator; add the numerators: Example: = We can represent this problem with a picture: Begin with the first fraction, 4 1, and add two more fourths ( 4 2 ): We now have threefourths of the whole shaded: 4 3 On Your Own: Add the following fractions = = = = Rule: When adding and subtracting fractions that have common denominators, we just add or subtract the numerators and keep the denominator. It gets trickier when the denominators are not the same. 16
21 Different Denominator Example: Let's represent this with a picture: The first picture shows one whole divided into four parts. One of these parts is shaded. We represent this as a fraction: 4 1 The second picture shows one whole divided into eight parts. Three of these parts are shaded. We represent this as a fraction: 8 3 In order to add these two fractions, we need to first divide them up into equal parts. The first picture is divided into fourths but the second is divided into eighths. We can easily convert the first picture into eighths by drawing two more lines (i.e. divide each fourth by half): 17
22 Now let's see how the first fraction would appear once it is divided into eighths: 1 2 We can see, from the above picture, that is equal to. 4 8 Now that we have a common denominator (8), we can add the 2 3 fractions: +. Just keep the denominator and add the 8 8 numerators: = Let's look at another example: Can we add these two fractions in their current form? Explain. To add two fractions, we need a common denominator. We must therefore convert the fractions to ones whose denominator is the same. We can use any common denominator, but it is much easier to use the lowest common denominator, or LCD. One way to find the LCD is to make a table and list, in order, the multiples of each denominator. (Multiple means Multiply!) 18
23 Finding the Lowest Common Denominator (LCD) Look at the last problem again: Now list the multiples of each denominator until you reach a common number. Multiples of 3 Multiples of The lowest common denominator (LCD) is the first common number in both columns: 15. This will be the new denominator for both fractions. Since we changed the denominators, we must also change the numerators so that each new fraction is equivalent (or equal) to the original fraction. Let s start with the first original fraction: 2/3. Go back to the table. How many times did we multiply the denominator, 3, by itself? (Hint: How many rows did we go down in the first column?) Since we multiplied the denominator (3) by to get 15, we must also multiply the numerator (2) by. Our new fraction is 15 Now let s look at the second fraction: 4/*5. Since we multiplied the denominator (5) by, we do the same to the numerator: 4 =. Our new fraction is 15 Now add the new fractions. + = We have an improper fraction because the numerator > the denominator. We must change it to a mixed number: 22 = 15 19
24 Let's look at another example: Example: Add the following fractions: In order to add these fractions we must first find a common denominator. Make a table and list all of the multiples for each denominator until we reach a common multiple: Multiples of 4 Multiples of We have a common denominator for both fractions: 20. Since we changed the denominators for both fractions, we must also change the numerators so that each new fraction is equivalent to the original fraction. Let s begin with the first fraction: 4 3 = 20 Now let s proceed to the second fraction: 5 4 = 20 Now both fractions have common denominators; add them: = 20 If the sum is an improper fraction (i.e. numerator > denominator), we generally change it to a proper fraction: 20
25 On Your Own Example: Step 1: Make a table and list the multiples of each denominator until you reach a common denominator: Step 2: Convert each fraction to an equivalent fraction: Step 3: Add the fractions: Note: If you end up with an improper fraction, be sure to convert it to a mixed number. 21
26 Practice = ' 7 3 = = = = =
27 Prime Factorization Another way to find the lowest common denominator of two fractions is through prime factorization. First, let s learn more about prime numbers: Prime Numbers: A prime number has two distinct whole number factors: 1 and itself. Note: 1 is not prime because it does not have two distinct factors. Example: 6 is not prime because it can be expressed as 2 3. Example: 7 is prime because it can be expressed only as the product of two distinct factors: 1 7. Write the first 10 prime numbers: Composite Numbers A nonprime number is called a composite number. Composite numbers can be broken down into products of prime numbers: Example: 4 = 2 X 2 Example: 12 = 2 X 6 = 2 X 2 X 3 Example: 66 = 6 X 11 = 2 X 3 X 11 Example: 24 = 2 X 12 = 2 X 2 X 2 X 3 Example: 33 = 3 X 11 Example: 125 = 5 X 5 X 5 23
28 Practice: Circle all of the prime numbers in the chart below:
29 Prime Factor Trees CAHSEE on Target We can find the prime factors of a number by making a factor tree: Example: Find the prime factors of 18. Write your number: 18 Begin with the smallest prime number factor of 18 (i.e. the smallest prime number that divides evenly 18. This number is 2. Draw two branches: 2 and the second factor: \ 2 9 Continue this process for each branch until you have no remaining composite numbers. The prime factors of 18 are the prime numbers at the ends of all the branches: 18 \ 2 9 \ 3 3 The prime factored form of 18 is. 25
30 Example: Find the prime factors of 60 using the factor tree: 60 \ 2 30 \ 2 15 \ 3 5 The prime factors of 60 are the factors at the end of each branch:,, and. Helpful Guidelines: Start with the smallest numbers: first 2 s, then 3 s, and so on. If a number is even, it is divisible by 2. Note: An even number ends in 0, 2, 4, 6, and 8. Examples: If the digits of a number add up to a number divisible by 3, the number is divisible by 3. Example: 123 can be divided evenly by 3 because if we add all of its digits, we get 6: = 6 Since the sum of the digits of 123 is divisible by 3, so too is 123. If a number ends in 0 or 5, it is divisible by 5. Examples:
31 On Your Own CAHSEE on Target Find the prime factors of each number, using a factor tree:
32 Prime Factorization and the Lowest Common Denominator On the CAHSEE, you will be asked to find the prime factored form of the lowest common denominator (LCD) of two fractions: Example: Find the prime factored form for the lowest common denominator There are two methods we can use to solve this problem: Method I: Factor Tree and Pairing Steps: Make a factor tree for both denominators: 6 9 \ \ Pair up common prime factors: Multiply the common factor (counted once) by all leftover (unpaired) factors: LCD = 3 = 28
33 Let's look at another example: Example: Find the least common multiple of 72 and 24. Write the LCM in primefactored form. Steps: Make a factor tree for each number: \ \ \ \ \ \ \ 3 3 Pair off common factors: 72 = = Count any common factor once! Multiply all common factors by all leftover (unpaired) factors: LCM = = 29
34 On Your Own: Solve the following problems, using the factor tree/pairing method. 1. What is the prime factored form of the lowest common denominator of ? 2. Find the least common multiple, in primefactorization form, of 12 and 15. We will now look at the second method to find the prime factored from of the lowest common denominator (LCD) of two fractions. 30
35 Method II: Factor Tree and Venn Diagram To illustrate this second method, let's return to the original problem: Example: Find the prime factored form for the lowest common denominator of Use the factor tree method to find the prime factored form of 6: 6 \ 2 3 Use the factor tree method to find the prime factored from of 9: 9 \ 3 3 Use a Venn diagram to find the primefactored form of the lowest common denominator: On the next page, we will learn how to fill out this diagram. 31
36 Venn Diagrams CAHSEE on Target Venn diagrams are overlapping circles that help us compare and contrast the characteristics of different things. We can use them to find what is common to two items (where the circles overlap in the middle) and what is different between them (what is outside the overlap on either or both sides). Here, we want to find out which prime factors are the same for two numbers and which factors are distinct, or different. 6 9 \ \ Steps: Since only one 3 is common to both numbers, we need to put it in the middle, where the two circles overlap: 6 Both 9 Continued on next page 32
37 Now find the prime factors that are left for 6 and place them in the part of the circle for 6 that does not overlap with the circle for 9. 6 Both 9 Next, find the prime factors that are left for 9 and place them in the part of the circle that does not overlap with the circle for 6. 6 Both 9 The lowest common denominator for 6 and 9 is the product of all of the numbers in the circles:, which is equal to Note: To write the LCD in primefactored form, we do not carry out the multiplication; we just write the prime numbers: LDC of 6 and 9 = 33
38 On Your Own 1. What is the prime factored form of the lowest common denominator of 6 1 and10 3? Create separate prime factor trees for both denominators: 6 10 \ \ Organize the prime factors of both denominators, using a Venn diagram: 6 Both 10 What is the LCD? Write the LCD in prime factored form: 34
39 2. Find the prime factored form of the lowest common denominator for the following: Factor Trees: 8 Both 12 LCD: LCD in prime factored form: 35
40 ii. Multiplying Fractions Whenever you are asked to find a fraction of a number, you need to multiply. In math, the word of means multiply. Example: Find 2 1 of 2 1. This is a multiplication problem. It means, What is ? We can represent the problem visually. Here is the first part of the problem: 2 1 of the circle has been shaded. Taking 2 1 of a number means dividing it by 2. Now, if we take onehalf of this again (divide it by 2 again), we get the following: of is equal to We end up with onefourth of the circle. Note: We also could have solved the above problem by multiplying the numerator by the numerator and the denominator by the denominator: Numerator Numerator_ = 1 1 = 1 Denominator Denominator
41 When working these problems out during the CAHSEE, you will need to apply this rule: Numerator Numerator Denominator Denominator Look at the next problem: Find 2 1 of 24. In math, we can write this as follows: The first factor is a fraction and the second factor is a whole number. We can easily change the second factor to a fraction because any whole number can be expressed as a fraction by placing it over a 1: = because 24 means 24 ones We can rewrite the problem as follows: 2 1 Now, just follow the rule for multiplying two fractions: Numerator Numerator Denominator Denominator = = Note: Taking 2 1 of 24 means dividing 24 by 2. 37
42 Now look at the next example: 24 5 Example: = 1 6 There are two ways to solve this problem: 1. The hard way: Perform all operations Multiply numerators: 24 5 Multiply denominators: 1 6 Divide new numerator by denominator: = 120 = = The easy way: Simplify first, and then multiply: _ = Simplify by dividing out common factors! Look at the following problems: = 3,435 = 79 = Do you need to work out these problems, or do you already know the answers? Remember: If you divide both a numerator and denominator by a common factor, you can make the problem much simpler to solve. So save yourself the time and work, and recognize these types of problems right away. 38
43 Look at the next set of problems: What do you notice about the above problems? There is a lot of heavy multiplication involved in these problems. Is there a way to make your work easier? Explain: We can fractions by before solving. We can simplify these problems quite a bit before solving. This makes our job easier. Let s look at the first problem: We can divide out common factors in each fraction. These common factors become clear if we write each fraction as a product of prime factors. Let's begin with the first fraction: 4 = = Now do the second fraction on your own: 6 = 8 Now let's multiply the two reduced fractions; but first, can we simplify anymore? If so, simplify first, and then multiply: 39
44 On Your Own: Simplify and solve: = = = = = = = =
45 iii. Dividing Fractions When you divide something by a fraction, think, How many times does the fraction go into the dividend? Example: dividend This means, How many times does go into? We can represent this visually: Answer: Example: We can represent this visually: Answer: 41
46 On Your Own: Solve the next few problems, asking each time, How many times does the fraction go into the whole number? = = = Do you see a pattern? Explain. 42
47 Reciprocals As we saw in the previous exercise, each time we divide a whole number by a fraction, we get as our answer the product of the whole number and the reciprocal of the fraction. Reciprocal means the flipside, or inverse. Example: The reciprocal of 5 4 is 4 5. On Your Own: Find the reciprocal of each fraction: Now let's find the reciprocal of a whole number. We know that any whole number (or integer) can be expressed as a fraction by placing it over 1: Example: 35 = 1 35 The reciprocal is the fraction turned upside down, or inverted: 1 Example: The reciprocal of 35 is 35 On Your Own: Find the reciprocal of each integer
48 Now we are ready to divide a whole number by a fraction Example: 2 = = = We can represent the above problem visually: means... If we count the number of little rectangles in the two big rectangles, we get. On Your Own = = = = = 44
49 Simplifying Division Problems Example: Remember the rule for dividing fractions: Rule: When dividing fractions, multiply the first fraction by the reciprocal of the second fraction! Steps: Multiplying the first fraction by the reciprocal of the second fraction, we get We can simplify this problem by dividing out common factors: Now, apply the rule for multiplication: Numerator Numerator = 1 2 = Denominator Denominator
50 On Your Own: Simplify and solve. 1 3 = = = = =
51 Unit Quiz: The following problems appeared on the CAHSEE ( ) = A. 3 1 B. 4 3 C. 6 5 D Which fraction is equivalent to ? 35 A. 48 B C D What is the prime factored form for the lowest common 2 7 denominator of the following: +12? 9 A. 3 X 2 X 2 B. 3 X 3 X 2 X 2 C. 3 X 3 X 3 X 2 X 2 D. 9 X 12 47
52 4. Which of the following is the prime factored form of the lowest 7 8 common denominator of +15? 10 A. 5 X 1 B. 2 X 3 X 5 C. 2 X 5 X 3 X 5 D. 10 X Which of the following numerical expressions results in a negative number? A. (7) + (3) B. (3) + (7) C. (3) + (7) D. (3) + (7) + (11) 6. One hundred is multiplied by a number between 0 and 1. The answer has to be. A. less than 0. B. between 0 and 50 but not 25. C. between 0 and 100 but not 50. D. between 0 and If x = 3, what is the value of x? A. 3 or 0 B. 3 or 3 C. 0 or 3 D. 9 or 9 48
53 8. What is the absolute value of 4? A. 4 B. C. 4 1 D The winning number in a contest was less than 50. It was a multiple of 3, 5, and 6. What was the number? A. 14 B. 15 C. 30 D. It cannot be determined 10. If n is any odd number, which of the following is true about n + 1? A. It is an odd number. B. It is an even number C. It is a prime number D. It is the same as n Which is the best estimate of 326 X 279? A. 900 B. 9,000 C. 90,000 D. 900,000 49
54 12. The table below shows the number of visitors to a natural history museum during a 4day period. Day Number of Visitors Friday 597 Saturday 1115 Sunday 1346 Monday 365 Which expression would give the BEST estimate of the total number of visitors during this period? A B C D John uses 3 2 of a cup of oats per serving to make oatmeal. How many cups of oats does he need to make 6 servings? A B 4 C D If a is a positive number and b is a negative number, which expression is always positive? A. a  b B. a + b C. a X b D. a b 50
55 Unit 2: Exponents On the CAHSEE, you will be given several problems on exponents. Exponents are a shorthand way of representing how many times a number is multiplied by itself. Example: can be expressed as 9 4 since four 9's are multiplied together. Base 9 4 exponent The number being multiplied is called the base. The exponent tells how many times the base is multiplied by itself. 9 4 is read as 9 to the 4 th power, or 9 to the power of 4. Let's look at another example: 2 = = 32 On Your Own 2³ = 2 = 3² = 3³ = Power of 0 Any number raised to the 0 power (except 0) is always equal to 1. Example: = 1 On Your Own 7 0 = = (131) 0 = 47 0 = 51
56 Power of 1 A number raised to the 1 st power (i.e., an exponent of 1) is always equal to that number. Example: = 100 On Your Own 7 1 = = (131) 1 = 47 1 = Power of 2 (Squares) A number raised to the 2 nd power is referred to as the square of a number. When we square a whole number, we multiply it by itself. Example: 12² = = 144 The square of any whole number is called a perfect square. Here are the first 3 perfect squares: 1² = 1 1 = 1 2² = 2 2 = 4 3² = 3 3 = 9 On Your Own: Write the perfect squares for the following numbers: 4² = 5² = 6² = 7² = 8² = 9² = 10² = 11² = 20² = (28)² (37)² = 3² + 5² = 52
57 Square Roots CAHSEE on Target The square root ( ) of a number is one of its two equal factors. Example: 8² = Any number raised to the second power (the power of 2) can be represented as a square. That s why it s called squaring the number. The square above has 64 units. Each side (the length and width) is 8 units. The area of the square is determined by multiplying the length (8 units) by the width (8 units). The square root is the number of units in each of the two equal sides: 8 Note: 64 has a second square root: 8 (88 = +64). However, when we are asked to evaluate an expression, we always take the positive root. Example: Find the square root of 36. Answer: 36 = 53
58 On Your Own = = = = = = = = = 10. Which is not a perfect square? A. 144 B. 100 C. 48 D
59 Power of 3 (Cubes) A number with an exponent of 3 (or a number raised to the 3 rd power) is the cube of a number. Example: 5³ = = 125 The cube of a whole number is called a perfect cube. Cubes of Positive Numbers The cube of a positive number will always be a positive number. 1³ = = 1 2³ = = 8 Cubes and Negative Numbers The cube of a negative number will always be a negative number. (1)³ = (1)(1)(1) = 1 (2) 3 = (2)(2)(2) = 8 On Your Own: Write the perfect cubes for the following numbers: 3³ = 4³ = 5³ = 3³ = 4³ = 5³ = 55
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