CONNECT: Powers and logs POWERS, INDICES, EXPONENTS, LOGARITHMS THEY ARE ALL THE SAME!

Save this PDF as:

Size: px
Start display at page:

Download "CONNECT: Powers and logs POWERS, INDICES, EXPONENTS, LOGARITHMS THEY ARE ALL THE SAME!"

Transcription

1 CONNECT: Powers and logs POWERS, INDICES, EXPONENTS, LOGARITHMS THEY ARE ALL THE SAME! You may have come across the terms powers, indices, exponents and logarithms. But what do they mean? The terms power(s), index (indices), exponent(s) in Mathematics are actually interchangeable. All of them are that little number written above, and to the right of, another number, such as 5 2 or 4 3. Some of those little numbers (written as superscripts) have special names. You are probably familiar with squaring and cubing a number. But let s start at the beginning! We might have to calculate 3 x 3. Rather than write out both those 3s, we use a shorthand notation: 3 2. The superscript 2 tells us that 3 is to be multiplied by itself, and we would get the answer 9. Note: the result is not 6 although there are two 3s, because two 3s would be 2 x 3, or 3 + 3, not 3 x 3. In the example 3 2, 3 is called the base and 2 is called the power (or index or exponent). We ll use power from now on, but remember that we can just as easily write index or exponent. 3 2 is read as three raised to the power of two, or simply three to the power two. More commonly, when the power is 2, we use the word squared, so we can also read this as three squared. No matter which way we express it, 3 2 will always mean 3 x 3 and give the answer 9. A further example: 4 3. This is read as four raised to the power of three, or four to the power three, or four cubed. It means 4 x 4 x 4 and will give the result 64 because 4 x 4 = 16 and 16 x 4 = 64. The base in this case is 4 and the power is 3. By the way, the powers 2 and 3 are the only ones that have special names. So, for example, 5 4 is read as five [raised] to the power [of] 4, or five [raised] to the 4 th [power]. Another example: 2 4 (read two to the power four) is 2 x 2 x 2 x 2, which makes 16. Here, the base is 2 and the power is 4. (Note how efficient the notation is we don t have to write out all those 2 s!) (Although it might seem trivial naming the base and power, they are important items of vocabulary for when we use logarithms we ll get to this later!) Over the page are some for you to try. 1

3 Operations with powers. Let s bring in a little bit of Algebra here. Now don t worry, Algebra simply generalises what happens to numbers. So, for example, a 2 just means a x a, where a is the base, 2 is the power and a can represent any number. There are some shortcuts to working out calculations with powers. Say we want to calculate a 2 x a 3. We could write this out longhand, and obtain a x a x a x a x a, which is a 5. Notice that the power, 5, is also the result of adding the powers 2 and 3. This happens in every case. So, to multiply two powers of the same base, just add the powers. This is our first general rule for operations with powers. We can use letters for the powers as well, but remember the letters simply stand for the general case and you can use the rule every time you recognise it. a p x a q = a p+q Example: Find the value of 2 5 x 2 4. Shortcut method: 2 5 x 2 4 = = 2 9. (Longer method: 2 5 x 2 4 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 2 9 ). Note that the base numbers must be the same for this rule to work. Now, we ve just seen that when you multiply the same base number raised to powers that you can actually add the powers. So it follows that if you are dividing the same base number raised to powers, then you would the powers 1. We can illustrate this as follows: = = = Did you think subtract? 3

4 This gives us our second general rule for operations with powers: a p a q = a p-q Write your answers to these questions using power notation: x x Combinations For example, (2 3 ) 4. This would mean 2 3 x 2 3 x 2 3 x 2 3. If we add the powers, we would end up with (And if we wrote out 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2, we would still get 2 12!) Notice we could have obtained the same result for the power if we had multiplied the 3 x 4. This is the same in every case. So, we can write: (a p ) q = a p q or we can also write (a p ) q = a pq (By the way, a number raised to the power 1 is just the number itself. Examples 3 1 = 3, = 752 etc.) Write your answer as a power in each case. 1. (3 2 ) 4 2. (2 5 ) 2 3. (4 1 ) 3 4. (10 2 ) 3 Zero power Taking the division rule a step further: what happens if we have ? Using what we know about squaring, 5 2 = 25, so we would get 25 25, which is 1. But what about using our other methods? 4

5 Firstly, using the subtraction of powers: = = 5 0 Now using division: = 1 So, 5 0 is the same as 1. This goes for any number, so we can generalise again and write: a 0 = 1 This is the 3 rd rule for indices/powers. Here are some for you to try: x (3 x 5) a 0 Negative powers And further than 0: let s say we have to calculate 3 4 subtraction of powers, (3 4-5 ), we would end up with Using the Using division, we would get = = 1 3 This means that 3-1 is the same as 1 3. If we had , we would end up with 3-4 or with 1 34, and this all leads to the rule : a p = 1 a p 5

6 Another example: 6-2 = = 1 36 Write each answer with a positive power: There is one more rule to deal with and we will do that on page 8. But first, let s discuss logarithms. Logarithms Although these really are the same as powers, we write them slightly differently. The easiest way to show you how logarithms work is by an example. (Note: we use log to stand for logarithm.) We know that 10 2 = 100. Now, let s try to put the 2 by itself and the 10 and the 100 on the other side of the equals sign. That s what logarithms do. A logarithm is a power. We write: log = 2. This is the shorthand that tells us that 2 is the power (or logarithm) to which we raise the base 10 to get the number 100. We say this as: the log[arithm] to base 10 of 100 is equal to 2. So 10 2 = 100 means exactly the same as log = 2. Another example: 2 5 = 32. So log 2 32 = 5, because 5 is the power to which we raise 2 to get 32. We read log to base 2 of 32 is equal to 5 (or sometimes log of 32 to base 2 is 5 ). Further example: 3-1 = 1 3. So, log 3( 1 3 ) = 1 Write each power statement in its log[arithm] form: = = = = 1 9 6

7 Now that you ve mastered power and log form, the next trick is to work out parts of logarithm expressions, such as the base, the number, or the log itself. For example, you might be asked to find the value of this log: log 2 8. So you need to work out the power to which 2 is raised to get 8. Solution: 2 x 2 x 2 = 8 so 2 3 = 8,which means that the log value is 3. So log 2 8 = 3. What about log ? We need to remember our place value, and that 0.1 is the same as 1, 10 which means the same as So, log is -1. You can actually check any log 10 on the calculator. Just type log 0.1 and you should see -1. (Note that on the calculator, log stands for log 10. You don t need to type the 10 into the calculator. Unfortunately, however, 10 is the only base for logs on the calculator, apart from the special case of logs to base e, which you may work with later.) Find the value of the log in each case: 1. log log log log log log 5 ( 1 25 ) 7. log 2 ( 1 8 ) 8. log 10( ) Now, find the value of the base. Let s call the base b, but only because we don t know its value yet. Here is an example: log b 9 = 2. So, we need to know what number squared is 9. That is, we need to find b for b 2 = 9. The base is 3 because 3 2 = 9, which means b = 3. (Notice that for b 2 = 9, b could also be 3, however we don t use negative bases with logs.) Find the value of the base (b). 1. log b 16 = 2 2. log b 81 = 4 3. log b = log b (½) = -1 7

8 Now, find the value of the number. Let s call the number n. Here is an example: log 2 n = 3. This means that 2 3 = n, so n = 8. Find the value of n. 1. log 2 n = 5 2. log 3 n = 4 3. log 2 n = log 4 n = -1 Last rule for powers Suppose we multiply 2 2. We must get 2, because that is what 2 means. In the same way, 3 3 = 3 or = 185. In fact we can even write a a = a, where a represents any number. But what about this? = = 2 1 = 2 Because we are multiplying by itself, and ended up with 2, just as I multiplied 2 by itself, and ended up with 2, it follows that 2 is the same as And 3 is the same as and so on. So we can write: 8

9 a 1 2 = a (Note: must not be confused with 2 1! The first is 2 raised to the power ½, 2 the second is read as two and a half.) As well, a = a (the cubed root of a), and a = a (the 4 th root of a). 3 (For example, 8 is 2 because we look for the number, which, when cubed gives 8. And so, = 2, for the same reason!) What about 8 2 3?! We can think of this in two different ways. Either, we can use (8 2 ) , which means we would look for 64 3 = 64 = 4, or we can use (8 1 3 ) 2 3, which means we would look for ( 8 ) 2 = 2 2 = 4. Either method works just as well (though the simpler numbers are probably easier). Another example: I m going to use ( ), because the numbers are simpler = 4 and 4 3 = 64. So = 64. (Notice we always use improper fractions for powers and do not use mixed numbers, so we use 3 instead of 1½.) 2 Again we can generalize this as a rule: p q a q = a p q or ( a) p Find the value of each of the following:

10 (You can check your answers on your calculator. If you are not sure how to do this, please refer to CONNECT: Calculators: GETTING TO KNOW YOUR SCIENTIFIC CALCULATOR.) Using fraction powers with logs This section puts it all together! Example: find the value of log 4 8. This means we need to know the power to which 4 is raised to get 8. Many people write the answer 2 here because 4 x 2 is 8. But remember, we are looking for 4 raised to a power, that is 4 x 4 x, not 4 x 2. Back to finding log 4 8. To what power can I raise 4, to get 8? This is a bit tough, because, if we do 4 2, we get 16, which is bigger than 8. But if we do 4 1, we only get 4. So our power must be somewhere between 1 and 2. What we can do is: let x be the value of the log we are looking for, that is, let x = log 4 8. If we put this into power form, we would get 4 x = 8. Now, here s the trick. 4 is the same as 2 2, and 8 is the same as 2 3. So, we have: (2 2 ) x = 2 3. We can multiply the powers and get 2 2x = is the same base on both sides, so the powers (2x and 3) must be the same, so 2x = 3. This tells us that x = 3 2. So, log 4 8 is 3 2. (You can check this by using your calculator, and finding if is 8.) Here is another example, with just the procedure set out. Find the value of log Let x = log x = 243 (3 2 ) x =

11 3 2x = 3 5 2x = 5 x = 5 2 log = 5 2 Last of all, there are some rules which we can look at for logarithms as well. We ll do them in CONNECT:Powers and logs2, where we will also look at some uses of logarithms. If you need help with any of the Maths covered in this resource (or any other Maths topics), you can make an appointment with Learning Development through Reception: phone (02) , or Level 3 (top floor), Building 11, or through your campus. 11

12 Answers (From page 2) is 8 (it is 2 x 2 x 2) = 81 (it is 3 x 3 x 3 x 3) = 100 (it is 10 x 10) = 125 (it is 5 x 5 x 5). Raising a negative number to a power (from page 2) 1. (-4) 2 = 16 (it is -4 x -4). 2. (-3) 4 = 81 (it is -3 x -3 x -3 x -3) = 10 x 10 x 10 5 x 5 x 5 = = (-5) 3 = 10 x 10 x x -5 x -5 = = 875 (note 3 and 4 are the same) = 10 x 10 4 x 4 = = 84 Operations with powers (from page 4) x 2 5 = = = = x = = 5 5 (or you can do = = = 5 5 ) Combinations (from page 4) 1. (3 2 ) 4 = (2 5 ) 2 = (4 1 ) 3 = (10 2 ) 3 = 10 6 Zero power (from page 5) = = x 3 0 = 4 x 1 = = = 5 3 or = = (3 x 5) 0 = 1 (you don t even have to worry about doing the inside of the brackets first here the 0 index tells you straight away that your result is a 0 means 4 x a 0 = 4 x 1 = 4 Negative powers(from page 6) = = = 4-3 =

13 Logarithms (from page 6) Writing in log form = 25 is the same as log 5 25 = = 64 is the same as log 4 64 = = 64 is the same as log 8 64 = = 1 9 is the same as log = 2 Finding the value of the log 1. log 2 4 = 2 because 2 2 = 4, so the answer (the logarithm, or power) is log = 3 because 10 3 = 1000, so the answer (the logarithm, or power) is log 3 81= 4 because 3 4 = 81, so the answer is log = -2, because 10-2 = 1 100, so the answer is log 5 25 = 2, because 5 2 = 25, so the answer is 2. 6.log 5 ( 1 ) = -2, because 5-2 = 1, so the answer is log 2 ( 1 ) 8 = -3, because 2-3 = 1, so the answer is log 10 ( 1 ) 1000 = -3, because 10-3 = 1, so the answer is Finding the base: 1. log b 16 = 2. This means b 2 = 16, so b = log b 81 = 4. This means b 2 = 81, so b = log b = -3. This means b 3 = 0.001, so b =

14 4. log b (½) = -1. This means b 1 = ½, so b = 2. Finding the number: 1. log 2 n = 5. This means 2 5 = n, so n = log 3 n = 4. This means 3 4 = n, so n = log 2 n = -3. This means 2-3 = n, so n = ⅛. 4. log 4 n = -1. This means 4-1 = n, so n = ¼. Last rule for powers (from page 9) = 25 = = 81 = = ( 49) 3 = 7 3 = = ( 125) 2 = 5 2 = 25 14

SOLVING EQUATIONS. Firstly, we map what has been done to the variable (in this case, xx):

CONNECT: Algebra SOLVING EQUATIONS An Algebraic equation is where two algebraic expressions are equal to each other. Finding the solution of the equation means finding the value(s) of the variable which

FROM THE SPECIFIC TO THE GENERAL

CONNECT: Algebra FROM THE SPECIFIC TO THE GENERAL How do you react when you see the word Algebra? Many people find the concept of Algebra difficult, so if you are one of them, please relax, as you have

Sometimes it is easier to leave a number written as an exponent. For example, it is much easier to write

4.0 Exponent Property Review First let s start with a review of what exponents are. Recall that 3 means taking four 3 s and multiplying them together. So we know that 3 3 3 3 381. You might also recall

Determining When an Expression Is Undefined

Determining When an Expression Is Undefined Connections Have you ever... Tried to use a calculator to divide by zero and gotten an error Tried to figure out the square root of a negative number Expressions

In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature.

Indices or Powers A knowledge of powers, or indices as they are often called, is essential for an understanding of most algebraic processes. In this section of text you will learn about powers and rules

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) =

Roots and Powers. Written by: Bette Kreuz Edited by: Science Learning Center Staff

Roots and Powers Written by: Bette Kreuz Edited by: Science Learning Center Staff The objectives for this module are to: 1. Raise exponential numbers to a power. 2. Extract the root of an exponential number.

OPERATIONS: x and. x 10 10

CONNECT: Decimals OPERATIONS: x and To be able to perform the usual operations (+,, x and ) using decimals, we need to remember what decimals are. To review this, please refer to CONNECT: Fractions Fractions

2 is the BASE 5 is the EXPONENT. Power Repeated Standard Multiplication. To evaluate a power means to find the answer in standard form.

Grade 9 Mathematics Unit : Powers and Exponent Rules Sec.1 What is a Power 5 is the BASE 5 is the EXPONENT The entire 5 is called a POWER. 5 = written as repeated multiplication. 5 = 3 written in standard

Solution Guide Chapter 14 Mixing Fractions, Decimals, and Percents Together

Solution Guide Chapter 4 Mixing Fractions, Decimals, and Percents Together Doing the Math from p. 80 2. 0.72 9 =? 0.08 To change it to decimal, we can tip it over and divide: 9 0.72 To make 0.72 into a

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

Section 1.2 Exponents and the Order of Operations

4. ( (4 + 1) ( + ) ) = ( 5 ( 8 )) = (5 5) = ( -51 ) = -15 Technical Writing 44. Explain different methods for grouping expressions and why it s important to have them. The student s answer should discuss

CONNECT: Calculations

CONNECT: Calculations ORDER OF OPERATIONS IN MATHEMATICS Operations in Mathematics are all those procedures like +,, x and, together with ( ), raising to a power and, which are used in calculating, and,

DECIMALS are special fractions whose denominators are powers of 10.

DECIMALS DECIMALS are special fractions whose denominators are powers of 10. Since decimals are special fractions, then all the rules we have already learned for fractions should work for decimals. The

Math 016. Materials With Exercises

Math 06 Materials With Exercises June 00, nd version TABLE OF CONTENTS Lesson Natural numbers; Operations on natural numbers: Multiplication by powers of 0; Opposite operations; Commutative Property of

4. Changing the Subject of a Formula

4. Changing the Subject of a Formula This booklet belongs to aquinas college maths dept. 2010 1 Aims and Objectives Changing the subject of a formula is the same process as solving equations, in that we

Exponents and Radicals (a + b) 10 Exponents are a very important part of algebra. An exponent is just a convenient way of writing repeated multiplications of the same number. Radicals involve the use of

Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 6 Math Circles November 4/5, 2014 Exponents Quick Warm-up Evaluate the following: 1. 4 + 4 + 4 +

FRACTIONS 1 MANIPULATING FRACTIONS. the denominator represents the kind of pieces the whole has been divided into

CONNECT: Fractions FRACTIONS 1 MANIPULATING FRACTIONS Firstly, let s think about what a fraction is. 1. One way to look at a fraction is as part of a whole. Fractions consist of a numerator and a denominator:

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

Decimal and Fraction Review Sheet

Decimal and Fraction Review Sheet Decimals -Addition To add 2 decimals, such as 3.25946 and 3.514253 we write them one over the other with the decimal point lined up like this 3.25946 +3.514253 If one

Solution: There are TWO square roots of 196, a positive number and a negative number. So, since and 14 2

5.7 Introduction to Square Roots The Square of a Number The number x is called the square of the number x. EX) 9 9 9 81, the number 81 is the square of the number 9. 4 4 4 16, the number 16 is the square

eday Lessons Mathematics Grade 8 Student Name:

eday Lessons Mathematics Grade 8 Student Name: Common Core State Standards- Expressions and Equations Work with radicals and integer exponents. 3. Use numbers expressed in the form of a single digit times

Grade 7/8 Math Circles October 7/8, Exponents and Roots - SOLUTIONS

Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 7/8 Math Circles October 7/8, 2014 Exponents and Roots - SOLUTIONS This file has all the missing

CONNECT: Algebra. 3x = 20 5 REARRANGING FORMULAE

CONNECT: Algebra REARRANGING FORMULAE Before you read this resource, you need to be familiar with how to solve equations. If you are not sure of the techniques involved in that topic, please refer to CONNECT:

CONNECT: Ways of writing numbers

CONNECT: Ways of writing numbers SCIENTIFIC NOTATION; SIGNIFICANT FIGURES; DECIMAL PLACES First, a short review of our Decimal (Base 10) System. This system is so efficient that we can use it to write

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

Word Problems. Simplifying Word Problems

Word Problems This sheet is designed as a review aid. If you have not previously studied this concept, or if after reviewing the contents you still don t pass, you should enroll in the appropriate math

(- 7) + 4 = (-9) = - 3 (- 3) + 7 = ( -3) = 2

WORKING WITH INTEGERS: 1. Adding Rules: Positive + Positive = Positive: 5 + 4 = 9 Negative + Negative = Negative: (- 7) + (- 2) = - 9 The sum of a negative and a positive number: First subtract: The answer

Arithmetic Circuits Addition, Subtraction, & Multiplication

Arithmetic Circuits Addition, Subtraction, & Multiplication The adder is another classic design example which we are obliged look at. Simple decimal arithmetic is something which we rarely give a second

Calculating Logarithms By Hand

Calculating Logarithms By Hand W. Blaine Dowler June 14, 010 Abstract This details methods by which we can calculate logarithms by hand. 1 Definition and Basic Properties A logarithm can be defined as

Introduction to the TI-84/83 Graphing Calculator

Introduction to the TI-84/83 Graphing Calculator Part 1: Basic Arithmetic Operations & Features 1. Turning the calculator ON & OFF, adjusting the contrast, and clearing the memory The ON key is located

( yields. Combining the terms in the numerator you arrive at the answer:

Algebra Skillbuilder Solutions: 1. Starting with, you ll need to find a common denominator to add/subtract the fractions. If you choose the common denominator 15, you can multiply each fraction by one

Free Pre-Algebra Lesson 24 page 1

Free Pre-Algebra Lesson page 1 Lesson Equations with Negatives You ve worked with equations for a while now, and including negative numbers doesn t really change any of the rules. Everything you ve already

Square roots by subtraction

Square roots by subtraction Frazer Jarvis When I was at school, my mathematics teacher showed me the following very strange method to work out square roots, using only subtraction, which is apparently

Maths Module 4. Powers, Roots and Logarithms. This module covers concepts such as: powers and index laws scientific notation roots logarithms

Maths Module 4 Powers, Roots and Logarithms This module covers concepts such as: powers and index laws scientific notation roots logarithms www.jcu.edu.au/students/learning-centre Module 4 Powers, Roots,

MBA Jump Start Program

MBA Jump Start Program Module 2: Mathematics Thomas Gilbert Mathematics Module Online Appendix: Basic Mathematical Concepts 2 1 The Number Spectrum Generally we depict numbers increasing from left to right

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

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

Welcome to Basic Math Skills!

Basic Math Skills Welcome to Basic Math Skills! Most students find the math sections to be the most difficult. Basic Math Skills was designed to give you a refresher on the basics of math. There are lots

Chapter 2 Formulas and Decimals

Chapter Formulas and Decimals Section A Rounding, Comparing, Adding and Subtracting Decimals Look at the following formulas. The first formula (P = A + B + C) is one we use to calculate perimeter of a

Solving Logarithmic Equations

Solving Logarithmic Equations Deciding How to Solve Logarithmic Equation When asked to solve a logarithmic equation such as log (x + 7) = or log (7x + ) = log (x + 9), the first thing we need to decide

COMPASS Numerical Skills/Pre-Algebra Preparation Guide. Introduction Operations with Integers Absolute Value of Numbers 13

COMPASS Numerical Skills/Pre-Algebra Preparation Guide Please note that the guide is for reference only and that it does not represent an exact match with the assessment content. The Assessment Centre

2. Simplify. College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key. Use the distributive property to remove the parentheses

College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key 1. Multiply 2 3 5 1 Use the distributive property to remove the parentheses 2 3 5 1 2 25 21 3 35 31 2 10 2 3 15 3 2 13 2 15 3 2

Practice Math Placement Exam

Practice Math Placement Exam The following are problems like those on the Mansfield University Math Placement Exam. You must pass this test or take MA 0090 before taking any mathematics courses. 1. What

HOSPITALITY Math Assessment Preparation Guide. Introduction Operations with Whole Numbers Operations with Integers 9

HOSPITALITY Math Assessment Preparation Guide Please note that the guide is for reference only and that it does not represent an exact match with the assessment content. The Assessment Centre at George

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

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.

Translating Mathematical Formulas Into Excel s Language

Translating Mathematical Formulas Into Excel s Language Introduction Microsoft Excel is a very powerful calculator; you can use it to compute a wide variety of mathematical expressions. Before exploring

Granby Primary School Year 5 & 6 Supporting your child with maths A handbook for year 5 & 6 parents H M Hopps 2016 G r a n b y P r i m a r y S c h o o l 1 P a g e Many parents want to help their children

47 Numerator Denominator

JH WEEKLIES ISSUE #22 2012-2013 Mathematics Fractions Mathematicians often have to deal with numbers that are not whole numbers (1, 2, 3 etc.). The preferred way to represent these partial numbers (rational

CHAPTER 3 Numbers and Numeral Systems

CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,

Chapter 15 Radical Expressions and Equations Notes

Chapter 15 Radical Expressions and Equations Notes 15.1 Introduction to Radical Expressions The symbol is called the square root and is defined as follows: a = c only if c = a Sample Problem: Simplify

GRE MATH REVIEW #5. 1. Variable: A letter that represents an unknown number.

GRE MATH REVIEW #5 Eponents and Radicals Many numbers can be epressed as the product of a number multiplied by itself a number of times. For eample, 16 can be epressed as. Another way to write this is

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

We could also take square roots of certain decimals nicely. For example, 0.36=0.6 or 0.09=0.3. However, we will limit ourselves to integers for now.

7.3 Evaluation of Roots Previously we used the square root to help us approximate irrational numbers. Now we will expand beyond just square roots and talk about cube roots as well. For both we will be

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

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

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

Basic numerical skills: POWERS AND LOGARITHMS

1. Introduction (easy) Basic numerical skills: POWERS AND LOGARITHMS Powers and logarithms provide a powerful way of representing large and small quantities, and performing complex calculations. Understanding

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

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

This explains why the mixed number equivalent to 7/3 is 2 + 1/3, also written 2

Chapter 28: Proper and Improper Fractions A fraction is called improper if the numerator is greater than the denominator For example, 7/ is improper because the numerator 7 is greater than the denominator

EXPONENTS. Judo Math Inc.

EXPONENTS Judo Math Inc. 8 th Grade Orange Belt Training Problem Solving Discipline Order of Mastery - Eponents (8.EE.1-4) 1. Foundations of Integer Eponents & Powers 2. Working & Simplifying Eponential

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

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

Section P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities

Section P.9 Notes Page P.9 Linear Inequalities and Absolute Value Inequalities Sometimes the answer to certain math problems is not just a single answer. Sometimes a range of answers might be the answer.

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

Grade 9 Mathematics Unit #1 Number Sense Sub-Unit #1 Rational Numbers. with Integers Divide Integers

Page1 Grade 9 Mathematics Unit #1 Number Sense Sub-Unit #1 Rational Numbers Lesson Topic I Can 1 Ordering & Adding Create a number line to order integers Integers Identify integers Add integers 2 Subtracting

Pre-Algebra Notes: , Absolute Value and Integers. Name: Block: Date: Z = {, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, }

Name: Block: Date: Section 2.1: Integers and Absolute Value We remember from our previous lesson that the set Z of integers looks like: Z = {, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, } Why do we care about

Domain of a Composition

Domain of a Composition Definition Given the function f and g, the composition of f with g is a function defined as (f g)() f(g()). The domain of f g is the set of all real numbers in the domain of g such

1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N.

CHAPTER 3: EXPONENTS AND POWER FUNCTIONS 1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N. For example: In general, if

Z-TRANSFORMS. 1 Introduction. 2 Review of Sequences

Z-TRANSFORMS. Introduction The Z-transform is a tranform for sequences. Just like the Laplace transform takes a function of t and replaces it with another function of an auxillary variable s, well, the

Mathematical Notation and Symbols

Mathematical Notation and Symbols 1.1 Introduction This introductory block reminds you of important notations and conventions used throughout engineering mathematics. We discuss the arithmetic of numbers,

1.1 THE REAL NUMBERS. section. The Integers. The Rational Numbers

2 (1 2) Chapter 1 Real Numbers and Their Properties 1.1 THE REAL NUMBERS In this section In arithmetic we use only positive numbers and zero, but in algebra we use negative numbers also. The numbers that

Numerical and Algebraic Fractions

Numerical and Algebraic Fractions Aquinas Maths Department Preparation for AS Maths This unit covers numerical and algebraic fractions. In A level, solutions often involve fractions and one of the Core

Rational Exponents. Given that extension, suppose that. Squaring both sides of the equation yields. a 2 (4 1/2 ) 2 a 2 4 (1/2)(2) a a 2 4 (2)

SECTION 0. Rational Exponents 0. OBJECTIVES. Define rational exponents. Simplify expressions with rational exponents. Estimate the value of an expression using a scientific calculator. Write expressions

Basic Use of the TI-84 Plus

Basic Use of the TI-84 Plus Topics: Key Board Sections Key Functions Screen Contrast Numerical Calculations Order of Operations Built-In Templates MATH menu Scientific Notation The key VS the (-) Key Navigation

Learning new things and building basic skills

Math Review TABE Answer Key 2 Learning new things and building basic skills may be challenging for you, but they also can be very exciting. When you follow the guidelines for learning basic skills, you

c sigma & CEMTL

c sigma & CEMTL Foreword The Regional Centre for Excellence in Mathematics Teaching and Learning (CEMTL) is collaboration between the Shannon Consortium Partners: University of Limerick, Institute of Technology,

Maths Module 6 Algebra Solving Equations This module covers concepts such as:

Maths Module 6 Algebra Solving Equations This module covers concepts such as: solving equations with variables on both sides multiples and factors factorising and expanding function notation www.jcu.edu.au/students/learning-centre

Year Five Maths Notes

Year Five Maths Notes NUMBER AND PLACE VALUE I can count forwards in steps of powers of 10 for any given number up to 1,000,000. I can count backwards insteps of powers of 10 for any given number up to

ACCUPLACER Arithmetic Assessment Preparation Guide

ACCUPLACER Arithmetic Assessment Preparation Guide Please note that the guide is for reference only and that it does not represent an exact match with the assessment content. The Assessment Centre at George

Absolute Value of Reasoning

About Illustrations: Illustrations of the Standards for Mathematical Practice (SMP) consist of several pieces, including a mathematics task, student dialogue, mathematical overview, teacher reflection

Numeracy Preparation Guide. for the. VETASSESS Test for Certificate IV in Nursing (Enrolled / Division 2 Nursing) course

Numeracy Preparation Guide for the VETASSESS Test for Certificate IV in Nursing (Enrolled / Division Nursing) course Introduction The Nursing course selection (or entrance) test used by various Registered

(Refer Slide Time: 00:00:56 min)

Numerical Methods and Computation Prof. S.R.K. Iyengar Department of Mathematics Indian Institute of Technology, Delhi Lecture No # 3 Solution of Nonlinear Algebraic Equations (Continued) (Refer Slide

Activity 1: Using base ten blocks to model operations on decimals

Rational Numbers 9: Decimal Form of Rational Numbers Objectives To use base ten blocks to model operations on decimal numbers To review the algorithms for addition, subtraction, multiplication and division

Rules for Exponents and the Reasons for Them

Print this page Chapter 6 Rules for Exponents and the Reasons for Them 6.1 INTEGER POWERS AND THE EXPONENT RULES Repeated addition can be expressed as a product. For example, Similarly, repeated multiplication

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

IB Math Research Problem

Vincent Chu Block F IB Math Research Problem The product of all factors of 2000 can be found using several methods. One of the methods I employed in the beginning is a primitive one I wrote a computer

+ = has become. has become. Maths in School. Fraction Calculations in School. by Kate Robinson

+ has become 0 Maths in School has become 0 Fraction Calculations in School by Kate Robinson Fractions Calculations in School Contents Introduction p. Simplifying fractions (cancelling down) p. Adding

Algebra 1A and 1B Summer Packet

Algebra 1A and 1B Summer Packet Name: Calculators are not allowed on the summer math packet. This packet is due the first week of school and will be counted as a grade. You will also be tested over the

Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 1 Section 8 Powers and Exponents

Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 1 Please watch Section 8 of this DVD before working these problems. The DVD is located at: http://www.mathtutordvd.com/products/item66.cfm

A Summary of Error Propagation

A Summary of Error Propagation Suppose you measure some quantities a, b, c,... with uncertainties δa, δb, δc,.... Now you want to calculate some other quantity Q which depends on a and b and so forth.

The notation above read as the nth root of the mth power of a, is a

Let s Reduce Radicals to Bare Bones! (Simplifying Radical Expressions) By Ana Marie R. Nobleza The notation above read as the nth root of the mth power of a, is a radical expression or simply radical.

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

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

R.5 Algebraic Expressions

Section R. Algebraic Expressions 39 R. Algebraic Expressions OBJECTIVES Translate English Expressions into the Language of Mathematics Evaluate Algebraic Expressions 3 Simplify Algebraic Expressions by

Module 2: Working with Fractions and Mixed Numbers. 2.1 Review of Fractions. 1. Understand Fractions on a Number Line

Module : Working with Fractions and Mixed Numbers.1 Review of Fractions 1. Understand Fractions on a Number Line Fractions are used to represent quantities between the whole numbers on a number line. A