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AlgebraNumbersBefore you can start anything in mathematicsare the basicsyou need to know how numbers work So hereThere is a summary on the next pageif you prefer flow diagramsNumbers are broken down into Real and Non"Real numbers Non"Real R’ numbers are numbers that don’t exist imaginary or complex numbersfor example the square"root of a negative number like 3 These numbers will give anerror on the calculator On the SHARP El",-, the calculator will sayError .Calculation Real Numbers R are numbers that do exist They are broken down into Rational andirrational numberso Irrational Numbers Q’ are numbers that don’t make sense I like to pretend theyare like my grandmothershe likes to talk and talk but she doesn’t make anysense In the same way2 irrational numbers go on and on called non"terminating and they don’t have a pattern or make sense called non"recurringAn example of an irrational number is 7 2,645751311 " do you see that itdoesn’t have a pattern and it doesn’t stop3o Rational numbers on the other hand are either recurring that means they have apattern for example 42------or they are terminating that means they endfor example .2., or - Rational numbers can be written as a fraction Rationalnumbers are also broken down into Whole numbers N4these are numbers that start at zero 4 and countup in wholes2 e g 42 72 .2 -etc They DO NOT have decimals orfractions A nice way to remember whole numbers start with zero is thatthere is an 4 in whOOle
Natural numbers Nthese are counting numbers which means thatthey start at one 7 and count up in wholes2 e g 72 .2 -etc They doNOT have decimals or fractions A nice way to remember that naturalnumbers start with one is that there is a 7 in Natura77Integers Z are positive and negative whole numbers2 e g42 72 .2 -"-2 ".2 "72etc They also do NOT have fractions or decimalsNumbersReal NumbersNon-real numbers Don’t existGive an error on yourcalculator.E.g. square-root of anegative numberWhole Numbers Start at zeroNo decimals orfractionsRational TerminatingOr recurringCan be written asa fraction. Like grannyNon-recurringNon-terminatingIntegersNatural Numbers Start at oneNo decimals orfractions: remember to learn both the names and the symbolssymbols or the namesIrrational Positive andnegative wholenumbersNo fractions ordecimals.because a question can use either the
Activity 77Tick the correct columns in the table below alInteger4"7"72;, 3 8 5.Given the equation 0 Solve forwhenarealbnon"realcan integer 3 2 5 6isRounding OffPart of knowing your numbers is remembering the facts about rounding offso here is somequick revision for you When you round off check how many places you want to round off to2 then look at the numbernext to the last number you will have once you have rounded off e g round off to three decimal places place rounding off to42 7.-;, number you use to decide whether thenumber next door goes up or stays the sameIf the number is , or bigger than , that is 2 2 ?2 @ then the number next door goes upIf the number is smaller than , that is ;2 -2.2 72 4 the number next door stays thesame
: However2 remember that when you are working with numbers representing people or thingsthat cannot be a fraction be careful how you round upthink carefully about what the questionis asking you for For example2 if you work out you have to cater for ;-2. people you wouldhave to round UP to ;; because you cannot not cater for the 42. part of a personAlsowhen they say round off to the nearest unitremember that a unit is a whole numberActivity .7.Round off the following numbers to . decimal places a427-,b.2- @c,2?@,d;2-,7Would you round the following up or down3athe average number of dogs per house is .2-bthe average number of people who live in a square meter in china is 772 cyour bill at the supermarket comes to R772@ dyou worked for .27- hoursethe average number of children in a class is ::::::::::::::::::Surds between IntegersA surd is a root of a number that cannot be simplified any further2 for example 3It is always good to estimate the answer of a surd so that you know that your calculatorcomputation was correctit’s a way of checking yourselfThe easiest way to check is to figure where the surd lies on the number line 3 1,732 which shows us that it lies between 7 and .But to work out where it lies without using a calculator you do this We count up in perfect squares 1 1; 2 4; 3 9; 4 16; 5 25 .etcFrom this we can see that - lies between 7 and ; on the perfect square line Then we reverse itto see that 3 lies between 7 and . the square"roots of the perfect square
Oklet’s practice that again Between which two integers does 18 lie3Count up in squares 72 ;2 @2 7 2 .,2 - We can see that 7? falls between 7 and .,Let’s take the square"root of 7 and ., to give us ; and ,2 so we know that 18 lies between; and ,Remember that if the question had said between 18 our answer would lie between;Activity -7Between which two integers do the following surds lie3aceg.: 56 78 43 99bdfh 12 15 29 8Between which two integers do the following surds lie3ac 6 78bd 45 ::::::::::::, and
ExponentsexponentBasepowerLet’s revise the basic laws of exponents Law 7 If you have the same bases and you multiply the power you add the exponents o " #"Law . If you have the same bases and you divide the powers2 you subtract theexponents o " %"Law - If you raise an exponent to another exponent you multiply the two exponentstogether "o" o Remember that all the powers in the bracket are affected by the outsideexponent "& '( "(&'( (Law ; If you take a root of a power the exponent inside is divided by the type of rooto *) *o A nice way to remember this rule is that cats live inside the house c 2 dogs liveoutside the house d and cats are more important than dogs so they go on top Law , Anything to the power of zero is oneo , 1Law If your power has a negative exponent the power goes under 7 and the exponentbecomes positive it also works in reverseif you have a power under 7 with a negativeexponent the power goes back on top and the exponent becomes positiveo%" -./0-1. -.but you can just write it as"o A nice way to remember this is to think of the exponent as the girlEboyfriend ofthe base2 when the exponent is negative the girlEboyfriend is negative and wantsto go downstairs or upstairs if the power is under the line Once the power isdownstairs or upstairs it makes the girlEboyfriend happy or positive
You need to learn these exponent laws as they will help you later on with other work that youwill cover all the way to matric Learn them now and you won’t be confused later There are two things you need to be able to do with exponents :solve:and simplifyTo simplify use your exponents to get to the simplest possible form of an answer make surethat all your exponents are positive 2 for example2 simplify 7-2 3) . - )435-26 3 ) 4First multiply top with top and bottom with bottom when you you GThen divide like bases when you H you ") 357 %8 7%& '7 %9& 'Finally write your answer with positive exponents-4 ) 2and now we can’t simplify anymore so this is our final answer3:9; 62 87 First break down the numbers into their prime factors7 1 ;@ I x I .? , I 7., x I , x , x , x I ,- x -, I , x 62827 87 8 1 Multiply out the exponents82 6: 7 8 7 1 8 1 7?#9#?% ?% 7 ?#9% ?# 5 77 5 84 035Simplify by adding when you multiply the bases and subtracting5?% ?%when dividing?% ?#To solve you will be given an equation and you need to work out what the value ofare three different ways of asking for :as an exponent:as a base:as an answeris There
Prime factors are factors of a number that are prime(they only have the factors 1 and itself). E.g. to primeHere are some examples factorise 48 – break 48 into factors – 6 x 8 and thenfurther break down those numbers into 3 x 2 and 4 x 2. 3and 2 are prime numbers so we can’t factorise themSolve for 73. 2 5 7?anymore. The only number is 4 which is 2 x 2. So 48 primeFirst takefactorised is 3x2x2x2x2 or 3 x 24the number that is not attached to the power to the other side3.2? 12Then divide by the -2? 4Now prime factorise the number2? 2Because the bases are the same we know the exponents are thesame2 so 2.5?# 5? 5 7 6 7 7 7 312 3125? @5 A 3125? @2 A 3125? 125-?9?9 ;Divide both sides by 2Drop the bases on both sides 3 4 2 699 27 27 347 Now take out the 5? as a common factorPrime factorise the 7.,5? 5 Separate the 5?# into two separate bases 1 024First take the number that is not attached to the power acrossNow multiply out by the ;Then take the cube"root of both sidesa cube"root of a cube cancels the other outSimply put the power into your calculator by pressing
Activity ;7Simplify the following a) :C2Dc1? :IE2 F 2 J 6 .egi.D G 1 bIJ 1d C 1 ) H: ? 4 1 ; .9;2 DG 1 :K 2 ℎ3? %O@IA fD H G : M2N? @I %DE 2 @ A PI?,%Ahj%3 @ A-12- . 9 6 362 1-1 3H' 6 F4-3: ) 4 ? %@? IAF 732-2 - 348 6 2 114 5Solve for acegi2? 106 2 6 7 0,3b 50 1 300 4?# 4? 5? 3.5? 4d9? :hj4? 7 21 5 103 4f7 6 2?# 9 1 17' 1 F32
Multiplying Algebraic ExpressionsWhen you multiply a single term with a bracket where there could be many terms each termin the bracket is multiplied by that single termWhen you have a bracket with more than one term2 being multiplied with a bracket that hasmore than one term you need to make sure that each term in the first bracket is multiplied byeach term in the second bracket An easy way to check that you are doing this is to drawarrows in one colour to each term in the second bracket2 then do the next term in anothercolour and so onFor example2 multiply 2 Q by 37.2 Q 3 4 Q Q; 4 Q Q-,Start with the 2 and multiply each term in the secondbracket with the 2arrows in blueterm2 Q arrows in green Then do the secondDon’t forget to multiply out thesigns as well766. 8-;Q 2 Q 3 11, Q 4 Q QNow simplify your answer by addingall the like terms togetherQ 6 Q QActivity ,Multiply out and simplify the following expressionsacegi3 5@9 1A @ 4 6 4 3 @7 4A 11 6A@9 ' 3FA @ ' 'F 6F A 2K 5ℎ 6K 6Kℎ ℎbdfhj@ Q 2A 4Q 6Q 36 4Q 2 4 Q 3 3 2& 7 8 & &5R 2E @R 8RE E A9@8S 7 TA @2S 9 S T STA
FactorisingFactorising is taking many terms and making them into one term Factorising will be usedthroughout the rest of your maths career so make sure you understand this sectionThere are four steps to factorisation 7Look for a highest common factor if there is one take it out2eg 8 the . is the highest common factor because it goes into both termsWhen you take it out in other words you divide each term by . and leave the . on theoutside of the brackets and the answers inside the brackets and put it in the front youget 2 4 Remember that if you multiply out the brackets you have just factorised itshould give you your original equation or expression. Now look at how many terms the expression hasa If there are two terms it could beiA difference of squares7 & sayTo factorise UEVWXY YRWT XR'Z[F YRWT UEVWXY YRWT XR'Z[F YRWTiiA difference or sum of cubes7\ & say To factorise UEVWXY YRWT \ XR'Z[F YRWT @ UEVWXY YRWT XR'Z[F YRWT XR'Z[F YRWT A UEVWXY YRWT b If there are three terms it could be a trinomialiThe easiest way to factorise a trinomial is to use your calculator2 forexample2 factorise 5 24 Some theory before I show you how remember after you have factorised you will have to bracketsmultiply them out you will get twoif youterms which you add whether thesign is minus or plus together to give you the middle term We are goingto use this fact in our methodTo find factor pairs go to table mode by pressing
Then enter your “c” or constant value e g ".; by typingthen press.;to put it at the top of a fraction At the bottom enter anX by pressingtwice Press thebuttonThis will take you through to a screen that says XNStart 4XNStep 7PresstwiceNow you will see a table with different factor pairsX4ANS""""""7" .;." 7.Look at your factor pairs and add each set together2 e g 7 G ".; I ".This is not our middle term so we look at the next factor pair . and "7.We continue until we find a factor pair that gives us ",2 in this case - and"? So now we know our brackets will beIf there is a number in front of your 3 8divide the entire trinomial by thatnumber or take it out as a common factorThen follow the same stepsabove but make your step a fraction 7 over the common factor you tookout2 6EgFactorise 7Take . out as a common factor .Go into table mode-For XNStart press;For XNStep press,Now look at the table and find the factors pairs that add up to 2@ 3Aand then type into leave it at 4
In our case the pair is 1[F 2 so the brackets will look like this ?Multiply your common factor back into the first bracket or the2@ 1 A 2bracket with the fraction to complete your factorising 2 3c 2If there are four terms it means you need to group terms that have the samevariable together and then take a common factor out of each group3eg 3 & 2 2&As you can seethis is already grouped together -’s together and .’stogetherTake a common factor from the first two terms -a and a common factor fromthe next two terms .3 & 2 &Now you can see that the brackets for each group are the same2 so we can takethese brackets out as a common factor and put the original common factors in abracket together behind the “new” common factor bracket & 3 2Activity 7Factorise the following by taking out a common factor ac.2 4 & 8&72 & ' 48&' 36 'b5' F 30' 9 F 85' 7 Fd9 Q 3b2Q 12Factorise the following cubes and squares acegi8 Q Q54 16216& 8d949R 64E Rfhj 2Q125 343K16' 25FQ Q9509 800Q 9
-Factorise the following trinomials aeikmo;4322 7 6d 12 2f 7 6h 19 5j 14 8l 11 21n 3 5p27 7 11 56 7 53 5 22 27 814 20 25Factorise the following by grouping acegi,3 9 20b 13 42cg 308 & 12& 20& 30 & 12 & 189& 20& 3016K 64K ℎ Kℎ 4ℎ' 18F 6'F 3' Fb7&2R E 6R E 3RE 9REd2 12S 3S 8 Sf3T 14[9 21T[ 2T [ 3 8 24 h7W 56W Y 10WY 80Ya 15a b 5a b 3bjFactorise the following use any of the above methodsacegikmoq280133610822suw316 49 25 500 Qd 30& 78 & 5& 6 24 42& 6 9 9 1109 Q96 48Q4 11 67j& 56 &npr&x& 125'0,01694 Q 2 43 65 81tv 16Q 9 1l 7 & 112& 16 7 40fh 5 4 3 18 6 9b9 24 Q 18 32 5 6 Q
Algebraic FractionsNow don’t skip this section just because the heading says algebraic fractions Fractions areactually pretty straight forward if you remember the basicsSo here are the basics Numerators are the numbers and variables at the top of the fraction2 denominators arethe numbers and variables at the bottom of the fraction"c Dd-efd CD"f g"-efdThe Golden rule What you do to the top e g multiply by two you do to the bottomalso multiply by two fairs fair Before you can add andEor subtract fractions2 you need to make sure the denominators the LCD lowest common denominator is How did we getare the same2 e gthat3 By multiplying the two denominators together . x - I In the same way for - the denominator is ab because a x b I ab3You can never ever cancel over two terms in other wordsdon’t cancel over a plus orminus If there are too many terms try factorising firstegeg? 2 #9 you cannot cancel the ’s because there are two terms on the top? 2 #9? first take out a common factor ofterm over one term? ?#9?Now you have oneremember the definition of factorising and so you can cancel soyour answer will now be at the top 4Remember to try to simplify before you add2 subtract2 multiply or dividethis will saveyou lots of time later on Your denominator can NEVER be zero 4This will cause your entire sum to beundefined Also remember that you can take out a negative 7 as a factor When you multiply fractions you multiply top with top and bottom with bottom When you are dividing fractions remember to turn the fraction up"side"down and thenmultiply2 e g-3)) ) - 3 -3
So let’s try some examples 7?# ?2%?#First factorise and simplify so that you can see all? 2 #?%the possible factors that could go into your commondenominator ?%?# ?#?%?#You can see that from the first fraction the?# 1 inthe numerator and the denominator cancel and the 2 in the numerator and the denominator of thesecond fraction cancel ?% ?%Now you can see that the denominators are thesame2 so you can add the numerators together ?%.? #I9?# I ?2?# I %?I#I 2?#I ? 2 %?I#I 2?#IAlways factorise first ?# I? 2 %?I#I 2Now see if you can cross cancel you can onlycross cancel if there is a multiply sign between thetwo fractions ?#I ? 2 %?I#I 2?#I ?# I? 2 %?I#I 2Do you see the three sets of “common” factors thatyou can take out3 ?# I Now multiply top with top and bottom with bottom 2 3Q--2 #-3 -3- -#3-3-2 %32 3Always factorise first-#3 -%33Remember that you cannot cross cancel across adivide sign so first turn the fraction up"side"down - -#3- 3- -#3-3 3-#3 -%33-#3 -%3Now you can cancelNotice that - goes into twice so there is aremainder of . at the top -%3 -%3Now you can multiply
;? %I? 2 %I 2 ?#I First factorise the expression?%I?%I ? 2 #?I#I 2?%I ?#I? 2 #?I#I 2?#I ?#I Cancel anything you can?%I?#IFind the LCD?%I?%I ? 2 #?I#I 2 # ?#I?#I?#I ?%I Qin this case Q and multiplyMultiply out and simplify? #? 2 I#?I 2 %I? 2 %?I 2 %I # ? 2 #?I# ?I#I 2?#I ?%I? # ? 2 # ?I#I 2 %IYou cannot factorise or simplify any further so your sum is?#I ?%IcompleteActivity Simplify the following fractions assume that all denominators do not equal zero acegi7? 2 % ,I 27?I((% i# ""-#3 % ?#Imj 2 #jk? %Ij 2 %k 2(%9i%9"-2 3-# 3?2Ikb7?# ,I(2 %9i 2 -2 %32? # I?I 2 #"%-3 -%3?# I? : %9 I : ?%I ? 2 k%jj#9k ?% I#?I#I 27j# k 9k#j d? % If? 2 %9h?# I--%3? 2 %9I 2? % -33% - ? 2 # ?I#9I 2?#%? ?# 3%- -#3 j-#3l? 2 #9?#9n-#3 ? #-2 %327 I-2 %32 -% 373%-% %?#7 -#3 -%3 32 %-29?# ::::::::::::
Solving EquationsWhat is the difference between an expression and an equation3An expression does not have an equal sign in the middleyou can only simplify an expression2whereas an equation has an equal sign in the middle and you can solve for a variable becausethe one side is equal to the other side2This is an expression And this is an equation Q 5 2 7Do you see that the expression can be simplified to 2 2Q and then cannot be changed ormanipulated anymore3Howeverwe can manipulate the equation so that we can actually find the value for 5 2 7 5 5 7 2 2 7 What you do to one side you do to the other side 7 7Do you remember that whatever you do to one side you HAVE to do to the other side3 FairsFairif your mom bought you an ice"cream but not your sister an ice"cream that wouldn’t befairin the same way you can’t do one thing on the one side and not do it on the other sidebecause that wouldn’t be very fair would it3 In fact you would actually be changing the equationand your final answer would be wrongThere are several different types of equations Linearwhen there are NO exponents in the equation2 e g 2 4 5o Its very easy to solve these kinds of equations2 you simply take all the ’s to oneside and the numbers to the other side and then find the value for oneo Eg2 4 5Take theto the left by subtracting it fromboth sides2 4 4 5 5 Now take the ; to the right 4 4 5 4 9When you are more confident you can dothe two steps together" just be careful not tomake a mistake
Quadratic & ' where a2 b2 and c arethese are equations in the formconstants numbers without variables or any equation where the highest power ofiso To solve quadratic equations you need to factorise the equation and make eachfactor equal to zero Then you solve for each factor o Eg 30First take everything over to the left so that you onlyhave a zero on the right 6 30 0Now you can factorise 5 0 Now we make each factor the part in bracketsequal to zero 6 0ZW 5 0Now solve forindividually 6 6 6 5 5 5 6 ZW 5o An easy way to remember that you need two answers is that there is a . on the Simultaneous Equationsthese are two equations that are happening at the same timeon the same Cartesian plane You have to figure out where the two “graphs” orequations will intersect In other words2 where are the two equations equal to eachother3 This means that the coordinates orand Q values will be the same for bothgraphs E equationso There are two methods for solving simultaneous equationsThe first is substitution you need to find either anor a Q by itself andthen substitute it into the other equation Then you will only have onevariable that you are solving for Once you find that variable2 substitute itback into the first equation to find the other variable’s value You canuse this in any kind of simultaneous equationsThe second method is elimination subtract the one equation from theother equation so that one of the variables is cancelled out You mayneed to multiply one of the equations first so that the one variable has thesame constant Then solve for the remaining variable Once you have theanswer you can substitute it back into one of the original equations to find
the value of the other variable use this method only if the two equationsare of the same typeo EgSolve forMethod 7 e g both are linear or both are quadraticand Q if Q 3 5 and 2Q 7 3SubstitutionGet eitherZW Q by itself already done for youNext Substitute it into the other equation Q 3 5 2 3 5 7 3Now solve that equation for66 10 7 3 10 10 7 7 3 7 10 13 13Now that you havesubstitute it back into Q 3 5 to find thevalue of yQ 3 13 5 Q 39 5 Q 44Method . EliminationIn this equation we will eliminate the Q first This means that weneed to multiply equation 7 by two so that the Q’s have the sameconstant2 Q 3 52Q 6 10Note the entire equation is multipliedNow subtract the one equation from the other it doesn’t matterwhich order you put them in2Q 6 102Q 7 30 13Now solve for 13 13Now we substitute it back into one of the original equations to findthe value of Q Q 3 13 5 Q 44
Word Problemsyes2 that means story sums All you have to do is read the storyvery carefully and underline all the important information Once you’ve done that put theinformation togethereither in linear equations or in a table format From there you canform the equations in order to solve for the variables They can use linear2 quadraticand simultaneous equations in word problems so pay special attention to how things aresaid Literal equationsthese are equations where you have to make one of the variablesthe subject of the formula in other words2 get that variable by itself and take all theother variables and numbers to the other sideo Egl Solve for r in terms of v2 h and tdNe 3l 3 l 3 dNedNFirst take the - to the other side 3 3el 3 Y Y l 3 WℎThen take the t to the other sidedNeY l 3 ℎ e m%N W YThen divide the rh by h to get r by itselfdNNNow you have r by itself so you have answered thequestion Inequalitiesthese involve P2 Q2 and signs They work exactly like an equals signwith two exceptions When you divide or times by a negative number your inequality sign flipsoverYou can never divide or multiply across the inequality sign by the variableyou are looking for because you do not know the sign whether itspositive or negative of the variableo When you are solving for an inequality you need to be able to draw a graph orrepresentation of the inequality on a number line Here are a couple of rules tofollow P Q are represented by open dots
are represented by closed dotsReal numbers are drawn with a solid lineIntegers are drawn with closed dots between the two values of the solvedinequalitydo you remember why3Whole numbers start from zero and are drawn with closed dots betweenthe two values of the solved inequalityNatural numbers start from one and are drawn with closed dots betweenthe two values of the solved inequalityo EgSolve for 2 3 5 0 and represent the answer graphically if 2 3 3 5 3Take the constant to the other side 2 2 ?% Divide by ". Remember because you are dividing%by a negative the inequality sign flips over 1";"-o Interval Notation"."747.-this is another way of writing inequalities Sometimes you willbe asked to give your answer in interval notationP Q are represented byand are represented by R and SPut the bracket first then the smallest number then “T” and then thebiggest number followed by the second bracketEg2 5 2; 5r
Activity ?7Solve fora2 1 9 6g3 5 i. 2 5 7 2f73h 2 4 2 3j 7 10 0cegi 5 3 9 7 3 2 42 7 44 025bf 8 0cdefghij3Q Q 3j 2Q 5 6Q 55Q 2 8Q 4 5Q 2 46Q andandand 1 03h 42 27 03Q 4 1 5 4 0d 13 15 0Q 2 3 12 36 0bSolve the following simultaneous equations fora; 5 1 5 1 4 8dSolve the following quadratic equations for a- 136 1 3 2b 10 4 4cein the following linear equations andandandandandandand2Q 5 82Q 16 0 5 2 0 16 8and Q 4Q 2 55Q 3Q 4 52Q 2 12Q 43 Q 2 8Q 27Q 5 Solve the following word problems aSarah is .4 years younger than her dad In , years’ time she will be half of herdad’s age How old are Sarah and her dad now3
bJoe cycles to school and back every day It takes him twice as long on the way toschool as on the way home If Joe lives 74km from the school2 determine howlong it takes Joe to go to school and back home if his average speed is .,kmEhcBob goes to the local café and buys - cokes and a packet of chips for R. Thenext time he goes to the café he buys . cokes and - packets of chips and itcosts him R., Work out how much a coke costs and a packet of chips costsdDan wants to work out how much fencing he needs for his farm He knows thatthe ground is a rectangular shape and that the breadth is equal to twice thelength plus 7km Determine the length of the rectangle and hence determine thelength of fencing Dan needs if the area of the farm is -44km.eGeorge knows that the perimeter of his fence is 744m around his rectangularpiece of garden2 and that the breadth is 5m Determine the length of thegarden given that the area is ;44m.fGiven the sketch of a wall below 5 3Determine the value s of x if the surface area of the wall is --m.gAnne knows that an ice"cream cone can hold ; 27.-@cm- of ice"cream If theformula for the volume of a cone is a W ℎ and Anne knows the height of thecone is ,cm2 determine how wide her scoop of ice"cream can be,For each of the following find the given variable in terms of the other variables acegFind r if a WFind i if s t 1 VFind h if a Find d if u W ℎb" [ 1 FdFind t if X Find r if a Ce9WfFind m if Q T 'hFindif Q 3
Determine the values forand represent the answers both graphically and in intervalnotationace3 4 5 82 5 55 6 2f 6 9 2 3 2h 8 2 5i 6 4d 4 1 2g 2 5 6b 2 3 10jAnswers for the ActivitiesActivity 77NumberRealNon"Non"RealRationalIrrational4"7"72;, 3 8 5.Given the equation 0 Solve forwhen 3 2 5isarealbnon"real can integer 3 ZW \ 6 3 7 6WholeNaturalInteger
Activity .7.a427-, 427;b.2- @ .2- c,2?@, ,2@4d;2-,7 ;2-,aupbupcdownddowneupActivity -7.Before we start we should do a “square” number line first7;a@7 .,- ;@ ;?7744between and ?bbetween "; and "-cbetween "@ and "?dbetween - and ;ebetween " and " fbetween " and ",gbetween @ and 74hbetween . and -Before we start we should do a “cube” number line first 7?. ;7.,.7 -;-abetween 7 and .bbetween "; and "-cbetween ; and ,dbetween " and ",
Activity ;7a) :C2DE2 F 2 D G 1 C 1 ) H:22) C C DD 1 G1 G2) :C:D G29% ) 2D G9#%E% %%3 @ A-12 C 1 ) 2C ' F R ' F7 R % E % b-2- -53 -232-2323) 2C4D 2G@ x . I -. x . c1? :IJ I ?: JHIJ 1: ? 4 Ie- 4 6 . 1 . ; .9;2 6 8.8. 2 1 2 .f82 2 8 6 6 .8 1 2 122 .8: #?% % 9? 7.3?%9? 1 82 33?#?% % ?%# ?% % ? 6 6 : 12 : 12?#?# % 9?%?# %9?#2%?32%? 7' 6 F4 6 .2 2 2 1 -1 3H @x ; I -. x .2 3 3 3 x @ I x -. 7?# 2 3?2J H% ?362 1 ?:I J H 7 . 9 6 ?: I J x .d? x- I .- x - 3H ) H-C :3H ) H-3: ) 4 F 7-H -3: ) 4 C 5-5 3 w )C :) w C H?# % 9?%- 34229?#-3: ) 4 C 5-C :-3 w ) C w2?#%9?#' 1 F3 -H 3 w )CC-H 3 w )
gDG 1 :K 2 ℎ3 i D : G1:M12 ND : M2 G: ND H G : M2N @D H G : M2ℎD H G : M2ND wG: M: D%AE 2hG12I#? 2 @D @ G2 N:I % @ A PI?O@IA 1,?II#? 2?ID x M:?I @DG2DGH N:? %? %@? IAj Zero is one 7A?IA @I#? 2A?2I2I 2 # I? 2 #? :8 6 2 1 Anything to the power ofI#? 2%A14 58 6 2 18. 148 6 2 18 14 14?# % ?%72 ?% % ?%7 7 7?# %?#7 2 ?% %?#7 7 2?#.Solve for a2? 6 7 2? 0,3 7 2? @1 A 7 2? @ A 7 2 ?b, 125 1250 125 5 219 2? @ A 21, 50 1 300 21 2? @2 A 21,10 10 6 2? 2 2? 16 2 ? 2% 1c2?# 2 ? 29 4d9? : 5 103 9? : 9 4 1089 324 81: 81 \3
6e 2 f 88,1874?# 4? g 4? 4 4? h7 4? 15 79 1 4? 2 4 ? 4% 2? 2 2 1 25? 3.5? 4i4? 4? @ A 1 4? 6,35 4? @1 A 17 4? 16 1 4 j 5? 1 3 4 5 4 4 5? 1 0 ? 1777 8 8 2 7Anything to the power of zero is oneActivity ,a3 5 3 3c 7@9 1A @ 9 9d 12 6 4Q 2 12 12 24 32 4 6b 18 5 20 30 38 30 1 4A 4 4 Q 3 4Q 18 8 Q 16 Q 12QQ 18 16 Q 12Q@ Q 2A 4Q 6Q 3 2Q 3Q Q 8Q 12Q 6 2Q 5Q 10 Q 6
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There are two things you need to be able to do with exponents : solve : and simplify To simplify use your exponents to get to the simplest possible form of an answer ˆmake sure that all your exponents are positive 2 for example2 simplify 7 -23 ) - )4 35 First multiply top wit
