Section 1.5 Exponents, Square Roots, and the Order of Operations

Size: px
Start display at page:

Transcription

1 Section 1.5 Exponents, Square Roots, and the Order of Operations Objectives In this section, you will learn to: To successfully complete this section, you need to understand: Identify perfect squares. Evaluating quantities (1.1) Use exponents to abbreviate The multiplication table (1.1) repeated multiplication. The definition of factors (1.1) Evaluate powers of 10. Find square roots of perfect squares. Evaluate expressions by applying the order of operations. INTRODUCTION Exponents and square roots are found in many formulas in math, chemistry, physics, economics, and statistics. The ability to perform calculations with exponents and square roots, for instance, helps astronomers provide a safe flight for the space shuttle and for the many satellites orbiting the earth. Research biologists often use exponents to explain the increase or decrease in the number of cells they are observing. When cells divide, split in two, the researcher can count twice as many cells as before. For example, Sheila is watching over the growth of a bacteria used in making yogurt, and she is taking notes on One cell at the start Two cells after one minute Four cells after two minutes... and so on. what she sees. Sheila notices that the cells double in number every minute. Here is a summary of her notes: Number of cells at start at 1 minute at 2 minutes at 3 minutes at 4 minutes at 5 minutes cell = 2 cells = 4 cells = 8 cells = 16 cells = 32 cells As part of her research, Sheila must make a prediction about the number of cells present at 8 minutes. Assuming the pattern will continue, Sheila notices that the number of cells can be predicted each minute by multiplying by another factor of 2: Number of cells at start At 1 minute At 2 minutes At 3 minutes At 4 minutes At 5 minutes Exponents, Square Roots, and the Order of Operations page 1.5 1

2 She realizes that at 3 minutes, there are three factors of 2, at 4 minutes there are four factors of 2, and so on. With this in mind, Sheila predicts that at 8 minutes there will be eight factors of 2 bacteria: = 256. As you might imagine, representing the number of bacteria by this repeated multiplication can become a bit tedious. Instead, there is a way that we can abbreviate repeated multiplication, and that is with an exponent. EXPONENTS Eight factors of 2 can be abbreviated as 2 8. The repeated factor, 2, is called the base, and the small, raised 8 is called the exponent or power. 2 8 is read as 2 to the eighth power. For values raised to the second power, such as 6 2, it s most common to say that the base is squared, such as 6 squared. The phrase 6 squared comes from the area of a square in which each side has a length of 6 units, as you will see later in this section. 2 8 is called the exponential form and is called the expanded form. An exponent indicates the number of factors of the base. Example 1: Write each in exponential form. a) b) c) 5 5 Procedure: To identify the exponent, count the number of factors of the base. Answer: a) 7 3 There are three factors of 7. b) 10 6 There are six factors of 10. c) 5 2 There are two factors of 5. YTI #1 Write each in exponential form.. Use Example 1 as a guide. a) b) c) Exponents, Square Roots, and the Order of Operations page 1.5 2

3 Example 2: Expand each and find its value. a) 2 3 b) 3 4 c) 7 2 Answer: a) 2 3 = = means three factors of 2. b) 3 4 = If you use the Associative Property to group two factors = (3 3) 3 3) at a time and then multiply them, you ll get 9 9 which = 9 9 = 81 is 81. c) 7 2 = 7 7 = means two factors of 7. YTI #2 Expand each and find its value. Use Example 2 as a guide. a) 6 2 b) 2 4 c) 9 3 How should we interpret 5 1? 5 1 means one factor of 5. In this case, there are no repeated factors of 5, there is only 5 itself. In other words, 5 1 = 5. This principle is true for any base: If b represents any base, then b 1 = b. YTI #3 Rewrite each without an exponent. a) 2 1 b) 9 1 c) 17 1 d) 1 1 THE POWERS OF 10 Numbers such as 100, 1,000 and 10,000 are called powers of 10 because 10 2 = = = = 1, = = 10,000 Based on the results above, notice that... The exponent of 10 indicates the number of zeros that follow the 1. Exponents, Square Roots, and the Order of Operations page 1.5 3

4 So, 10 6 is a 1 followed by six zeros: 1,000,000; in other words, 1,000,000 is the 6th power of 10. Likewise, 10,000,000 is a 1 followed by seven zeros and is abbreviated as Example 3: For each power of 10, write its value or abbreviate it with an exponent, whichever is missing. a) 1,000 b) 100,000 c) 10 4 d) 10 8 Procedure: For a) and b), count the number of zeros; for c) and d), the exponent indicates the number of zeros following a 1. Answer: a) 1,000 = 10 3 b) 100,000 = 10 5 c) 10 4 = 10,000 c) 10 8 = 100,000,000 YTI #4 For each power of 10, write its value or abbreviate it with an exponent, whichever is missing. Use Example 3 as a guide. a) 10 5 b) 10 7 c) 10 1 d) 100 e) 10,000 f) 1,000,000 Three hundred can be written as 300 and as This can be further abbreviated as In other words, 300 = When a number ends in one or more zeros, it can be written as a product of a whole number and a power of 10. For example, 45,000,000 has six zeros following the 45, so the sixth power of 10, or 10 6, is a factor: 45,000,000 = Example 4: Identify the power of 10, and rewrite the number as a product of a whole number and a power of 10. a) 6,000 b) 290,000 c) 506,000,000 d) 80 Procedure: Count the number of zeros following the non-zero digit(s); that is the power of 10. Answer: a) 6,000 = followed by three zeros. b) 290,000 = followed by four zeros. c) 506,000,000 = followed by six zeros. d) 80 = followed by one zero. Exponents, Square Roots, and the Order of Operations page 1.5 4

5 YTI #5 As outlined in Example 5, rewrite the number as a product of a whole number and a power of 10. a) 240 = b) 5,600 = c) 308,000,000 = d) 7,260,000 = PERFECT SQUARES This is a diagram of a unit square, just as we saw in Section unit 1 unit In the squares below, notice that there is a number associated with each the number of unit squares within. Each number is called a perfect square because the unit squares form a square. 2 units 3 units 2 units 3 units 2 units x 2 units = 4 square units 3 units x 3 units = 9 square units 4 is a perfect square. 9 is a perfect square. Whenever we create a square with a number, n, as the length of one side, that same number, n, must appear as the length of the other side. The product of these two numbers, n n, is called a perfect square number, or just a perfect square. n n In other words, because 1 1 = 1 2 = 1, 1 is a perfect square 2 2 = 2 2 = 4, 4 is a perfect square 3 3 = 3 2 = 9, 9 is a perfect square 4 4 = 4 2 = 16, 16 is a perfect square 5 5 = 5 2 = 25, 25 is a perfect square The list of perfect squares goes on and on. Any time you multiply a whole number by itself you get a result that is (automatically) a perfect square. Exponents, Square Roots, and the Order of Operations page 1.5 5

6 Notice, also, that there are a lot of numbers that are not perfect squares: 2, 3, 5, 6, 7,... and the list goes on and on. 2 is not a perfect square because no whole number multiplied by itself equals 2. The same is true for all the numbers in this list. Example 5: Is it possible to draw a square that has: a) 81 unit squares within it? b) 24 unit squares within it? Answer: a) Yes, a square with 9 units on each side will have 81 square units within it because 9 9 = 81. b) No, there is no whole number that, when multiplied by itself, will give 24. YTI #6 Use Example 5 as a guide to answer the following questions. Is it possible to draw a square that has: a) 36 unit squares within it? b) 12 unit squares within it? c) 49 unit squares within it? Example 6: Find the perfect square number associated with each product. a) 7 2 b) 11 2 c) 13 2 d) 20 2 Procedure: Expand each and evaluate. Answer: a) 7 7 = 49 c) = 169 b) = 121 d) = 400 YTI #7 Find the perfect square number associated with each product. Use Example 6 as a guide. a) 6 2 b) 10 2 c) 18 2 Exponents, Square Roots, and the Order of Operations page 1.5 6

7 SQUARE ROOTS Consider this perfect square of 16. We know from our discussion of perfect squares, the number 4 makes the perfect square 16. For this reason, we call 4 a square root of units 4 units The Square Root Property If p = r 2 (p is a perfect square using r as one side), then r is a square root of p. Example 7: What is a square root of the following numbers? a) 49 b) 36 c) 1 d) 25 Procedure: What number, multiplied by itself, gives each of those above? Answer: a) A square root of 49 is 7. b) A square root of 36 is 6. c) A square root of 1 is 1. d) A square root of 25 is 5. YTI #8 What is a square root of the following numbers? Use Example 7 as a guide. a) A square root of 4 is. b) A square root of 9 is. c) A square root of 64 is. d) A square root of 100 is. THE SQUARE ROOT SYMBOL The square root symbol,, called a radical, makes it easier to write square roots. So, instead of writing A square root of 16 is 4, we can simply write 16 = 4. The number within the radical, in this case 16, is called the radicand. Exponents, Square Roots, and the Order of Operations page 1.5 7

8 When we use the radical to find a square root of a number, like 16, the radical becomes an operation. Just like the plus sign tells us to apply addition to two numbers, the radical tells us to take the square root of a number. Example 8: Evaluate each square root. a) 1 b) 25 c) 144 Procedure: Because each radicand, here, is a perfect square, applying the radical leaves a number with no radical symbol. Answer: a) 1 = 1 b) 25 = 5 c) 144 = 12 YTI #9 Evaluate each square root. Use Example 8 as a guide. a) 4 = b) 9 = c) 121 = d) 36 = e) 100 = f) 81 = THE ORDER OF OPERATIONS Let s review the operations we ve worked with so far: Operation Example Name Multiplication 3 5 = 15 product Division 14 2 = 7 quotient Subtraction 12 9 = 3 difference Addition = 10 sum Exponent 2 3 = 8 power Radical 25 = 5 square root As we learned earlier, to evaluate a mathematical expression means to find the value of the expression. Exponents, Square Roots, and the Order of Operations page 1.5 8

9 Consider this expression: How could it be evaluated? If we apply the operation of addition first, and then multiplication, we get = 7 5 = 35 If, however, we evaluate using multiplication first and then addition, we get = = 23 Notice that, depending on which operation is applied first, we get two different results. Math, however, is an exact science and doesn t allow for two different values of the same expression. Because of this, a set of guidelines, called the order of operations, was established. The order of operations tells us which operation should be applied first in an expression. The order of operations was developed with these thoughts in mind: 1. Any quantity within grouping symbols should be applied first. Grouping symbols includes parentheses ( ), brackets [ ], braces { }, and the radical, or the square root symbol,. 2. Because an exponent is an abbreviation for repeated multiplication, exponents should rank higher than multiplication. 3. Because multiplication and division are inverse operations they should be ranked together. Also, because multiplication is an abbreviation for repeated addition (and division is an abbreviation for repeated subtraction), multiplication (and division) should rank higher than addition (and subtraction). 4. Because addition and subtraction are inverse operations they should be ranked together. There are basically two types of grouping symbols: Those that form a quantity, like ( ), [ ], and { }, and those that are actual operations, like. The radical is both a grouping symbol and an operation! In summary, Exponents, Square Roots, and the Order of Operations page 1.5 9

10 The Order Of Operations 1. Evaluate within all grouping symbols (one at a time), if there are any. 2. Apply any exponents. 3. Apply multiplication and division reading from left to right. 4. Apply addition and subtraction reading from left to right. We sometimes refer to the order of the operations by their rank. For example, we might say that an exponent has a higher rank than multiplication. Because multiplication and division have the same rank, we must apply them (carefully) from left to right. You ll see how we do this in the following examples. Think about it: When evaluating an expression, is it ever possible to apply addition before multiplication? Explain. The best way to understand these guidelines is to put them to work. We ll find that there is only one way to evaluate an expression using the rules, but we ll also find that some steps can be combined in certain situations. For now, though, let s evaluate each expression one step at a time. Example 9: Evaluate each according to the order of operations. a) b) (14 6) 2 c) d) (7 + 3) 2 e) f) 24 (4 2) Answer: Each of these has two operations; some have grouping symbols that will affect the order in which the operations are applied. a) Two operations, subtraction and division: divide first. = 14 3 Notice that the minus sign appears in the second step. That s because it hasn t been applied yet. = 11 Exponents, Square Roots, and the Order of Operations page

11 b) (14 6) 2 Here are the same two operations as above, this time with grouping symbols. Evaluate the expression inside the parentheses first. = 8 2 Because we ve already evaluated within the grouping symbols, we don t need the parentheses any more. = 4 c) Two operations, addition and exponent: apply the exponent first, then add. = = 16 d) (7 + 3) 2 The same two operations as in (c); work within the grouping symbols first. = (10) 2 (10) 2 is also 10 2 ; at this point, the parentheses are no longer necessary. = 100 e) Two operations: division and multiplication. Because they have the same rank, and there are no grouping symbols to tell us which = 6 2 to apply first, we apply them in order from left to right. That means that division is applied first. = 12 f) 24 (4 2) This time we do have grouping symbols, so we begin by evaluating the expression within the parentheses. = 24 8 = 3 YTI #10 Evaluate each according to the order of operations. First identify the two operations, then identify which is to be applied first. Show all steps! Use Example 9 as a guide. a) b) 24 (6 + 2) c) d) (10 3) 2 e) f) (12 2) 2 Exponents, Square Roots, and the Order of Operations page

12 Some expressions contain more than two operations. In those situations, we need to be even more careful when we apply the order of operations. Example 10: Evaluate each according to the order of operations. a) b) 36 (3 6) 2 c) 36 [3 (6 2)] Answer: Each of these has three operations. In part c) there is a smaller quantity (6 2) within the larger quantity of the brackets, [ ]. a) Because multiplication and division have the same rank, we apply them from left to right. We divide first. = Notice that we are applying only one operation at a time = 72 2 and rewriting everything else = 70 b) 36 (3 6) 2 Evaluate the expression within the grouping symbols first. = Divide. = 2 2 Subtract. = 0 c) 36 [3 (6 2)] Start with what is inside the large brackets. Inside those grouping symbols is another quantity, and we must evaluate it first: 6 2 = 4. = 36 [3 4] Evaluate within the brackets: 3 4 = 12. = Divide. = 3 Example 10(c) illustrates that when one quantity is within another one, the inner quantity is to be evaluated first. Exponents, Square Roots, and the Order of Operations page

13 Sometimes an expression will have two sets of grouping symbols that are unrelated to each other, meaning that evaluating one does not affect the evaluation of the other. In other words, some quantities can be evaluated at the same time. For example, in the expression regardless of the operation: (8 3) (12 4) we can evaluate within each grouping symbol (8 3) (12 4) Here there are three operations: subtraction, multiplication and division. = (5) (3) Subtraction and division can be applied at the same time because the order of operations = 15 tells us to begin by evaluating what is inside of the grouping symbols. YTI #11 Evaluate each according to the order of operations. Identify the order in which the operations should be applied. On each line, write the operation that should be applied and then apply it. Use Example 10 as a guide. a) b) 36 (3 + 3) 2 c) 36 ( ) d) e) (11 + 4) (6 1) f) 11 + [4 (6 1)] g) (6 + 3) h) (2 3) 2 (6 + 3) Now let s look at some examples that contain a radical. Exponents, Square Roots, and the Order of Operations page

14 Example 11: Evaluate each completely. Remember, the radical is both a grouping symbol and an operation. a) b) c) Procedure: Because the radical is a grouping symbol, we must evaluate within it first. Answer: a) First apply addition. = 16 Now apply the square root. = 4 b) Apply both exponents within the same step. = Next apply addition. = 25 Apply the square root. = 5 c) The radical is a grouping symbol, so we should apply it = the square root first. Next, apply multiplication and then = 13 6 subtraction. = 7 YTI #12 Evaluate each according to the order of operations. Identify the order in which the operations should be applied. Use Example 11 as a guide. a) 4 9 b) 25 9 c) 1 + (12 4) d) (6 2) 5 2 Calculator Tip: Scientific Calculator Exponents, Square Roots, and the Order of Operations page

15 Is your calculator a non-scientific calculator or a scientific calculator? Input the expression x 5 to find out. A non-scientific calculator is programmed to evaluate an expression in the order that the operations are put into it. On this calculator, each time you push an operation key (or the = key) the expression is evaluated, whether you are finished or not. For example, on a non-scientific calculator, the expression x 5 would be evaluated in this way: Key sequence: x 5 = Display: We have 3 so far = 7 7 x 5 = 35 A scientific calculator, though, is programmed to always use the order of operations correctly. In doing so, it is patient and waits see what operation button is push next before deciding what to evaluate. For example, on a scientific calculator, the expression x 5 would be evaluated in this way: Key sequence: x 5 = Display: 23. To determine if your calculator a scientific calculator or not, input the expression x 5 and see which answer you get. You Try It Answers: Section 1.5 Exponents, Square Roots, and the Order of Operations page

16 YTI #1: a) 4 7 b) 3 4 c) 1 8 YTI #2: a) 6 6 = 36 b) = 16 c) = 729 YTI #3: a) 2 b) 9 c) 17 d) 1 YTI #4: a) 100,000 b) 10,000,000 c) 10 d) 10 2 e) 10 4 f) 10 6 YTI #5: a) b) c) d) YTI #6: a) Yes, a square with 6 units on each side will have 36 square units. b) No, there is no whole number that, when multiplied by itself, equals 12. c) Yes, a square with 7 units on each side will have 49 square units. YTI #7: a) 36 b) 100 c) 324 YTI #8: a) 2 b) 3 c) 8 d) 10 YTI #9: a) 2 b) 3 c) 11 d) 6 e) 10 f) 9 YTI #10: a) 6 b) 3 c) 4 d) 14 e) 3 f) 36 YTI #11: a) 18 b) 12 c) 4 d) 34 e) 75 f) 31 g) 2 h) 4 YTI #12: a) 6 b) 2 c) 7 d) 10 Expand each and find its value. Focus Exercises: Section 1.5 Exponents, Square Roots, and the Order of Operations page

17 = = = = = = = = = = Express each as a power of ,000 = 12. 1,000,000,000 = ,000 = 14. 1,000,000 = Rewrite each number as a product of a whole number and a power of = 16. 5,000 = ,000 = ,000 = 19. 9,500,000 = ,000,000 = Evaluate the following square roots Evaluate each according to the order of operations. Simplify just one step, one operation at a time (5 + 1) 31. (8 + 5) 2 Exponents, Square Roots, and the Order of Operations page

18 (6 3) (7 2) (2 3) (5 2) (5 + 3) (6 + 12) (2 3) [12 (2 3)] [28 (7 3)] (6 2) ( ) (7 3) 54. [(6 2) 3] (6 2) (24 8) (22 2) (8 + 2) (4 5) [9 (3 + 1)] 2 Exponents, Square Roots, and the Order of Operations page

19 74. [9 + (3 1)] (5 2) (5 2) A square has a side length of 4 yards. 78. A square has a side length of 9 inches. a) Draw and label all four sides of a) Draw and label all four sides of the square. the square. b) Find the perimeter of the square. b) Find the perimeter of the square. c) Find the area of the square. c) Find the area of the square. 79. The floor of a cottage is in the shape 80. A city park is in the shape of a square. of a square. Each side is 20 feet long. Each side is 100 feet long. a) Draw and label all four sides of a) Draw and label all four sides of the square. the square. b) Find the perimeter of the square. b) Find the perimeter of the square. c) Find the area of the square. c) Find the area of the square. 81. A square has an area of 36 square feet. 82. A square has an area of 100 square inches. a) What is the length of each side a) What is the length of each side of the square? of the square? b) What is the perimeter of the square? b) What is the perimeter of the square? 83. A table top is in the shape of a square and 84. A sand box is in the shape of a square and has an area of 9 square feet. has an area of 49 square yards. a) What is the length of each side a) What is the length of each side of the table top? of the sand box? b) What is the perimeter of the table top? b) What is the perimeter of the sand box? Exponents, Square Roots, and the Order of Operations page

1.6 The Order of Operations

1.6 The Order of Operations Contents: Operations Grouping Symbols The Order of Operations Exponents and Negative Numbers Negative Square Roots Square Root of a Negative Number Order of Operations and Negative

Direct Translation is the process of translating English words and phrases into numbers, mathematical symbols, expressions, and equations.

Section 1 Mathematics has a language all its own. In order to be able to solve many types of word problems, we need to be able to translate the English Language into Math Language. is the process of translating

Section 4.1 Rules of Exponents

Section 4.1 Rules of Exponents THE MEANING OF THE EXPONENT The exponent is an abbreviation for repeated multiplication. The repeated number is called a factor. x n means n factors of x. The exponent tells

ARE YOU A RADICAL OR JUST A SQUARE ROOT? EXAMPLES

ARE YOU A RADICAL OR JUST A SQUARE ROOT? EXAMPLES 1. Squaring a number means using that number as a factor two times. 8 8(8) 64 (-8) (-8)(-8) 64 Make sure students realize that x means (x ), not (-x).

Order of Operations More Essential Practice

Order of Operations More Essential Practice We will be simplifying expressions using the order of operations in this section. Automatic Skill: Order of operations needs to become an automatic skill. Failure

5.1 Radical Notation and Rational Exponents

Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots

Exponential Notation and the Order of Operations

1.7 Exponential Notation and the Order of Operations 1.7 OBJECTIVES 1. Use exponent notation 2. Evaluate expressions containing powers of whole numbers 3. Know the order of operations 4. Evaluate expressions

Exponents. Exponents tell us how many times to multiply a base number by itself.

Exponents Exponents tell us how many times to multiply a base number by itself. Exponential form: 5 4 exponent base number Expanded form: 5 5 5 5 25 5 5 125 5 625 To use a calculator: put in the base number,

HFCC Math Lab Beginning Algebra 13 TRANSLATING ENGLISH INTO ALGEBRA: WORDS, PHRASE, SENTENCES

HFCC Math Lab Beginning Algebra 1 TRANSLATING ENGLISH INTO ALGEBRA: WORDS, PHRASE, SENTENCES Before being able to solve word problems in algebra, you must be able to change words, phrases, and sentences

A Quick Algebra Review

1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals

No Solution Equations Let s look at the following equation: 2 +3=2 +7

5.4 Solving Equations with Infinite or No Solutions So far we have looked at equations where there is exactly one solution. It is possible to have more than solution in other types of equations that are

Rational Exponents. Squaring both sides of the equation yields. and to be consistent, we must have

8.6 Rational Exponents 8.6 OBJECTIVES 1. Define rational exponents 2. Simplify expressions containing rational exponents 3. Use a calculator to estimate the value of an expression containing rational exponents

23. RATIONAL EXPONENTS

23. RATIONAL EXPONENTS renaming radicals rational numbers writing radicals with rational exponents When serious work needs to be done with radicals, they are usually changed to a name that uses exponents,

PREPARATION FOR MATH TESTING at CityLab Academy

PREPARATION FOR MATH TESTING at CityLab Academy compiled by Gloria Vachino, M.S. Refresh your math skills with a MATH REVIEW and find out if you are ready for the math entrance test by taking a PRE-TEST

Solving Linear Equations in One Variable. Worked Examples

Solving Linear Equations in One Variable Worked Examples Solve the equation 30 x 1 22x Solve the equation 30 x 1 22x Our goal is to isolate the x on one side. We ll do that by adding (or subtracting) quantities

Quick Reference ebook

This file is distributed FREE OF CHARGE by the publisher Quick Reference Handbooks and the author. Quick Reference ebook Click on Contents or Index in the left panel to locate a topic. The math facts listed

This is a square root. The number under the radical is 9. (An asterisk * means multiply.)

Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize

Decimal Notations for Fractions Number and Operations Fractions /4.NF

Decimal Notations for Fractions Number and Operations Fractions /4.NF Domain: Cluster: Standard: 4.NF Number and Operations Fractions Understand decimal notation for fractions, and compare decimal fractions.

Determine If An Equation Represents a Function

Question : What is a linear function? The term linear function consists of two parts: linear and function. To understand what these terms mean together, we must first understand what a function is. The

Session 29 Scientific Notation and Laws of Exponents. If you have ever taken a Chemistry class, you may have encountered the following numbers:

Session 9 Scientific Notation and Laws of Exponents If you have ever taken a Chemistry class, you may have encountered the following numbers: There are approximately 60,4,79,00,000,000,000,000 molecules

All the examples in this worksheet and all the answers to questions are available as answer sheets or videos.

BIRKBECK MATHS SUPPORT www.mathsupport.wordpress.com Numbers 3 In this section we will look at - improper fractions and mixed fractions - multiplying and dividing fractions - what decimals mean and exponents

CAHSEE on Target UC Davis, School and University Partnerships

UC Davis, School and University Partnerships CAHSEE on Target Mathematics Curriculum Published by The University of California, Davis, School/University Partnerships Program 006 Director Sarah R. Martinez,

3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.

Session 7 Fractions and Decimals

Key Terms in This Session Session 7 Fractions and Decimals Previously Introduced prime number rational numbers New in This Session period repeating decimal terminating decimal Introduction In this session,

Grade 7/8 Math Circles Sequences and Series

Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 7/8 Math Circles Sequences and Series November 30, 2012 What are sequences? A sequence is an ordered

3.1. RATIONAL EXPRESSIONS

3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers

Math Matters: Why Do I Need To Know This? 1 Probability and counting Lottery likelihoods

Math Matters: Why Do I Need To Know This? Bruce Kessler, Department of Mathematics Western Kentucky University Episode Four 1 Probability and counting Lottery likelihoods Objective: To demonstrate the

A.2. Exponents and Radicals. Integer Exponents. What you should learn. Exponential Notation. Why you should learn it. Properties of Exponents

Appendix A. Exponents and Radicals A11 A. Exponents and Radicals What you should learn Use properties of exponents. Use scientific notation to represent real numbers. Use properties of radicals. Simplify

SIMPLIFYING SQUARE ROOTS

40 (8-8) Chapter 8 Powers and Roots 8. SIMPLIFYING SQUARE ROOTS In this section Using the Product Rule Rationalizing the Denominator Simplified Form of a Square Root In Section 8. you learned to simplify

Simplifying Square-Root Radicals Containing Perfect Square Factors

DETAILED SOLUTIONS AND CONCEPTS - OPERATIONS ON IRRATIONAL NUMBERS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you!

SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS

(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic

Unit 7 The Number System: Multiplying and Dividing Integers

Unit 7 The Number System: Multiplying and Dividing Integers Introduction In this unit, students will multiply and divide integers, and multiply positive and negative fractions by integers. Students will

SUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills

SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)

Pre-Algebra - Order of Operations

0.3 Pre-Algebra - Order of Operations Objective: Evaluate expressions using the order of operations, including the use of absolute value. When simplifying expressions it is important that we simplify them

8-6 Radical Expressions and Rational Exponents. Warm Up Lesson Presentation Lesson Quiz

8-6 Radical Expressions and Rational Exponents Warm Up Lesson Presentation Lesson Quiz Holt Algebra ALgebra2 2 Warm Up Simplify each expression. 1. 7 3 7 2 16,807 2. 11 8 11 6 121 3. (3 2 ) 3 729 4. 5.

Lesson 4. Factors and Multiples. Objectives

Student Name: Date: Contact Person Name: Phone Number: Lesson 4 Factors and Multiples Objectives Understand what factors and multiples are Write a number as a product of its prime factors Find the greatest

MATH 60 NOTEBOOK CERTIFICATIONS

MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5

0.8 Rational Expressions and Equations

96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions - that is, algebraic fractions - and equations which contain them. The reader is encouraged to

General Exponent Rules: Exponents, Radicals, and Scientific Notation x m x n = x m+n Example 1: x 5 x = x 5+ = x 7 (x m ) n = x mn Example : (x 5 ) = x 5 = x 10 (x m y n ) p = x mp y np Example : (x) =

Rules of Exponents. Math at Work: Motorcycle Customization OUTLINE CHAPTER

Rules of Exponents CHAPTER 5 Math at Work: Motorcycle Customization OUTLINE Study Strategies: Taking Math Tests 5. Basic Rules of Exponents Part A: The Product Rule and Power Rules Part B: Combining the

Verbal Phrases to Algebraic Expressions

Student Name: Date: Contact Person Name: Phone Number: Lesson 13 Verbal Phrases to s Objectives Translate verbal phrases into algebraic expressions Solve word problems by translating sentences into equations

A positive exponent means repeated multiplication. A negative exponent means the opposite of repeated multiplication, which is repeated

Eponents Dealing with positive and negative eponents and simplifying epressions dealing with them is simply a matter of remembering what the definition of an eponent is. division. A positive eponent means

SOLVING EQUATIONS WITH RADICALS AND EXPONENTS 9.5. section ( 3 5 3 2 )( 3 25 3 10 3 4 ). The Odd-Root Property

498 (9 3) Chapter 9 Radicals and Rational Exponents Replace the question mark by an expression that makes the equation correct. Equations involving variables are to be identities. 75. 6 76. 3?? 1 77. 1

Factoring Numbers. Factoring numbers means that we break numbers down into the other whole numbers that multiply

Factoring Numbers Author/Creation: Pamela Dorr, September 2010. Summary: Describes two methods to help students determine the factors of a number. Learning Objectives: To define prime number and composite

Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 1 Section 9 Order of Operations

Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 1 Please watch Section 9 of this DVD before working these problems. The DVD is located at: http://www.mathtutordvd.com/products/item66.cfm

2.3 Solving Equations Containing Fractions and Decimals

2. Solving Equations Containing Fractions and Decimals Objectives In this section, you will learn to: To successfully complete this section, you need to understand: Solve equations containing fractions

Multiplying and Dividing Signed Numbers. Finding the Product of Two Signed Numbers. (a) (3)( 4) ( 4) ( 4) ( 4) 12 (b) (4)( 5) ( 5) ( 5) ( 5) ( 5) 20

SECTION.4 Multiplying and Dividing Signed Numbers.4 OBJECTIVES 1. Multiply signed numbers 2. Use the commutative property of multiplication 3. Use the associative property of multiplication 4. Divide signed

Chapter 7 - Roots, Radicals, and Complex Numbers

Math 233 - Spring 2009 Chapter 7 - Roots, Radicals, and Complex Numbers 7.1 Roots and Radicals 7.1.1 Notation and Terminology In the expression x the is called the radical sign. The expression under the

Everything you wanted to know about using Hexadecimal and Octal Numbers in Visual Basic 6

Everything you wanted to know about using Hexadecimal and Octal Numbers in Visual Basic 6 Number Systems No course on programming would be complete without a discussion of the Hexadecimal (Hex) number

CALCULATIONS & STATISTICS

CALCULATIONS & STATISTICS CALCULATION OF SCORES Conversion of 1-5 scale to 0-100 scores When you look at your report, you will notice that the scores are reported on a 0-100 scale, even though respondents

Click on the links below to jump directly to the relevant section

Click on the links below to jump directly to the relevant section What is algebra? Operations with algebraic terms Mathematical properties of real numbers Order of operations What is Algebra? Algebra is

Unit 1 Number Sense. In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions.

Unit 1 Number Sense In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions. BLM Three Types of Percent Problems (p L-34) is a summary BLM for the material

Toothpick Squares: An Introduction to Formulas

Unit IX Activity 1 Toothpick Squares: An Introduction to Formulas O V E R V I E W Rows of squares are formed with toothpicks. The relationship between the number of squares in a row and the number of toothpicks

QM0113 BASIC MATHEMATICS I (ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION)

SUBCOURSE QM0113 EDITION A BASIC MATHEMATICS I (ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION) BASIC MATHEMATICS I (ADDITION, SUBTRACTION, MULTIPLICATION AND DIVISION) Subcourse Number QM 0113 EDITION

How do you compare numbers? On a number line, larger numbers are to the right and smaller numbers are to the left.

The verbal answers to all of the following questions should be memorized before completion of pre-algebra. Answers that are not memorized will hinder your ability to succeed in algebra 1. Number Basics

Chapter 1: Order of Operations, Fractions & Percents

HOSP 1107 (Business Math) Learning Centre Chapter 1: Order of Operations, Fractions & Percents ORDER OF OPERATIONS When finding the value of an expression, the operations must be carried out in a certain

Decimals and other fractions

Chapter 2 Decimals and other fractions How to deal with the bits and pieces When drugs come from the manufacturer they are in doses to suit most adult patients. However, many of your patients will be very

Mathematics Placement

Mathematics Placement The ACT COMPASS math test is a self-adaptive test, which potentially tests students within four different levels of math including pre-algebra, algebra, college algebra, and trigonometry.

Decimals are absolutely amazing We have only 10 symbols, yet can represent any number, large or small We use zero (0) as a place holder to allow us

Decimals 1 Decimals are absolutely amazing We have only 10 symbols, yet can represent any number, large or small We use zero (0) as a place holder to allow us to do this 2 Some Older Number Systems 3 Can

Fractions Packet. Contents

Fractions Packet Contents Intro to Fractions.. page Reducing Fractions.. page Ordering Fractions page Multiplication and Division of Fractions page Addition and Subtraction of Fractions.. page Answer Keys..

Summer Assignment for incoming Fairhope Middle School 7 th grade Advanced Math Students

Summer Assignment for incoming Fairhope Middle School 7 th grade Advanced Math Students Studies show that most students lose about two months of math abilities over the summer when they do not engage in

Integer Operations. Overview. Grade 7 Mathematics, Quarter 1, Unit 1.1. Number of Instructional Days: 15 (1 day = 45 minutes) Essential Questions

Grade 7 Mathematics, Quarter 1, Unit 1.1 Integer Operations Overview Number of Instructional Days: 15 (1 day = 45 minutes) Content to Be Learned Describe situations in which opposites combine to make zero.

Lies My Calculator and Computer Told Me

Lies My Calculator and Computer Told Me 2 LIES MY CALCULATOR AND COMPUTER TOLD ME Lies My Calculator and Computer Told Me See Section.4 for a discussion of graphing calculators and computers with graphing

MATH-0910 Review Concepts (Haugen)

Unit 1 Whole Numbers and Fractions MATH-0910 Review Concepts (Haugen) Exam 1 Sections 1.5, 1.6, 1.7, 1.8, 2.1, 2.2, 2.3, 2.4, and 2.5 Dividing Whole Numbers Equivalent ways of expressing division: a b,

OA3-10 Patterns in Addition Tables Pages 60 63 Standards: 3.OA.D.9 Goals: Students will identify and describe various patterns in addition tables. Prior Knowledge Required: Can add two numbers within 20

Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.

Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}

Session 6 Number Theory

Key Terms in This Session Session 6 Number Theory Previously Introduced counting numbers factor factor tree prime number New in This Session composite number greatest common factor least common multiple

Student Outcomes Students understand that the sum of two square roots (or two cube roots) is not equal to the square root (or cube root) of their sum. Students convert expressions to simplest radical form.

Creating Basic Excel Formulas

Creating Basic Excel Formulas Formulas are equations that perform calculations on values in your worksheet. Depending on how you build a formula in Excel will determine if the answer to your formula automatically

Integers are positive and negative whole numbers, that is they are; {... 3, 2, 1,0,1,2,3...}. The dots mean they continue in that pattern.

INTEGERS Integers are positive and negative whole numbers, that is they are; {... 3, 2, 1,0,1,2,3...}. The dots mean they continue in that pattern. Like all number sets, integers were invented to describe

REVIEW SHEETS BASIC MATHEMATICS MATH 010

REVIEW SHEETS BASIC MATHEMATICS MATH 010 A Summary of Concepts Needed to be Successful in Mathematics The following sheets list the key concepts that are taught in the specified math course. The sheets

NF5-12 Flexibility with Equivalent Fractions and Pages 110 112

NF5- Flexibility with Equivalent Fractions and Pages 0 Lowest Terms STANDARDS preparation for 5.NF.A., 5.NF.A. Goals Students will equivalent fractions using division and reduce fractions to lowest terms.

Year 9 set 1 Mathematics notes, to accompany the 9H book.

Part 1: Year 9 set 1 Mathematics notes, to accompany the 9H book. equations 1. (p.1), 1.6 (p. 44), 4.6 (p.196) sequences 3. (p.115) Pupils use the Elmwood Press Essential Maths book by David Raymer (9H

Introduction to Fractions

Section 0.6 Contents: Vocabulary of Fractions A Fraction as division Undefined Values First Rules of Fractions Equivalent Fractions Building Up Fractions VOCABULARY OF FRACTIONS Simplifying Fractions Multiplying

Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving

Section 7 Algebraic Manipulations and Solving Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving Before launching into the mathematics, let s take a moment to talk about the words

Time needed. Before the lesson Assessment task:

Formative Assessment Lesson Materials Alpha Version Beads Under the Cloud Mathematical goals This lesson unit is intended to help you assess how well students are able to identify patterns (both linear

Algebra I Teacher Notes Expressions, Equations, and Formulas Review

Big Ideas Write and evaluate algebraic expressions Use expressions to write equations and inequalities Solve equations Represent functions as verbal rules, equations, tables and graphs Review these concepts

Problem of the Month: Double Down

Problem of the Month: Double Down The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core

Algebra 1. Practice Workbook with Examples. McDougal Littell. Concepts and Skills

McDougal Littell Algebra 1 Concepts and Skills Larson Boswell Kanold Stiff Practice Workbook with Examples The Practice Workbook provides additional practice with worked-out examples for every lesson.

Properties of Real Numbers

16 Chapter P Prerequisites P.2 Properties of Real Numbers What you should learn: Identify and use the basic properties of real numbers Develop and use additional properties of real numbers Why you should

Veterans Upward Bound Algebra I Concepts - Honors

Veterans Upward Bound Algebra I Concepts - Honors Brenda Meery Kaitlyn Spong Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) www.ck12.org Chapter 6. Factoring CHAPTER

Vieta s Formulas and the Identity Theorem

Vieta s Formulas and the Identity Theorem This worksheet will work through the material from our class on 3/21/2013 with some examples that should help you with the homework The topic of our discussion

MATH 10034 Fundamental Mathematics IV

MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.

Measurements 1. BIRKBECK MATHS SUPPORT www.mathsupport.wordpress.com. In this section we will look at. Helping you practice. Online Quizzes and Videos

BIRKBECK MATHS SUPPORT www.mathsupport.wordpress.com Measurements 1 In this section we will look at - Examples of everyday measurement - Some units we use to take measurements - Symbols for units and converting

A Prime Investigation with 7, 11, and 13

. Objective To investigate the divisibility of 7, 11, and 13, and discover the divisibility characteristics of certain six-digit numbers A c t i v i t y 3 Materials TI-73 calculator A Prime Investigation

POLYNOMIAL FUNCTIONS

POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a

MULTIPLICATION AND DIVISION OF REAL NUMBERS In this section we will complete the study of the four basic operations with real numbers.

1.4 Multiplication and (1-25) 25 In this section Multiplication of Real Numbers Division by Zero helpful hint The product of two numbers with like signs is positive, but the product of three numbers with

Indices and Surds. The Laws on Indices. 1. Multiplication: Mgr. ubomíra Tomková

Indices and Surds The term indices refers to the power to which a number is raised. Thus x is a number with an index of. People prefer the phrase "x to the power of ". Term surds is not often used, instead

Week 13 Trigonometric Form of Complex Numbers

Week Trigonometric Form of Complex Numbers Overview In this week of the course, which is the last week if you are not going to take calculus, we will look at how Trigonometry can sometimes help in working

Basic Formulas in Excel. Why use cell names in formulas instead of actual numbers?

Understanding formulas Basic Formulas in Excel Formulas are placed into cells whenever you want Excel to add, subtract, multiply, divide or do other mathematical calculations. The formula should be placed

A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions

A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25

If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C?

Problem 3 If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Suggested Questions to ask students about Problem 3 The key to this question

Find the Square Root

verview Math Concepts Materials Students who understand the basic concept of square roots learn how to evaluate expressions and equations that have expressions and equations TI-30XS MultiView rational

1.6 Division of Whole Numbers

1.6 Division of Whole Numbers 1.6 OBJECTIVES 1. Use repeated subtraction to divide whole numbers 2. Check the results of a division problem 3. Divide whole numbers using long division 4. Estimate a quotient

Arithmetic Review ORDER OF OPERATIONS WITH WHOLE NUMBERS

Arithmetic Review The arithmetic portion of the Accuplacer Placement test consists of seventeen multiple choice questions. These questions will measure skills in computation of whole numbers, fractions,

7 Literal Equations and

CHAPTER 7 Literal Equations and Inequalities Chapter Outline 7.1 LITERAL EQUATIONS 7.2 INEQUALITIES 7.3 INEQUALITIES USING MULTIPLICATION AND DIVISION 7.4 MULTI-STEP INEQUALITIES 113 7.1. Literal Equations