1 BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook - Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly be review material.) a Fidig Square Roots. Recall: Whe we raise a umber to the secod power, c, we say it is squared. The square of a umber is the umber times itself. I symbols, a a a. For example, The square of is because. The square of is ( ) ( )( ). Whe we wat to fid what umber was squared, we are fidig a square root (the iverse of squarig). The iverse of squarig is fidig a square root. For example, Oe square root of is because. The other square root of is because ( ), too. Squarig ad fidig a square root are iverse operatios. Every positive umber will have two real-umber square roots, (oe positive ad oe egative). The umber 0 (zero) has just oe square root, 0 itself. Example Fid the square roots of. The square roots of are ad because ad ( ). Example Fid the square root of 0. The oly square root of 0 is 0 (sice 0 is ot positive or egative, so those choices do t exist here). Now Do Practice Exercises.. Squarig ad fidig a square root are operatios. Fid all square roots for each of the followig Radical Notatio: a The pieces of a radical expressio for square roots: a, are the radical sig or radical:, ad the radicad, a. (NOTE: The radicad is the etire expressio uder the radical, eve if it s composed of several terms or factors).
2 Radical Notatio: If a > 0 the a b, b > 0 ad if b a. I words, if a is a positive umber, a is equal to the positive umber b whose square is a. a is the positive square root or pricipal square root of a ad Whe the radical ( ) sig is used, to ask for the egative square root, a egative sig must be writte i frot of the radical: a b, b >0 ad if b a Also ote: 0 0. Example Fid each square root. a. 9 b. Solutio: a. 9 because 9. b. because.. a is read the square root of a. Example Fid: Solutio: because Now Do Practice Exercises The radicad of 8 is.. The symbol,, is called a. Fid each square root b Approximatig Square Roots. So far, we ve looked at square roots of perfect squares. Numbers like ¼, 6,, ad are called perfect squares because they re the square of a whole umber (or a fractio) ad their square root is ratioal (see part C below for the formal defiitio of ratioal). A square root such as caot be writte as a whole umber sice is t a perfect square. Such umbers are called irratioal (see part C below for the formal defiitio of irratioal). Although ca t be writte as a whole umber, it ca be approximated betwee two earest wholes by fidig the perfect squares surroudig the radicad or it ca be approximated to about decimal places usig a calculator. Sice is betwee ad 9, is betwee ad 9, i.e., is betwee ad. Usig a calculator, Cautio: these are approximatios ad are ot the exact value of. The exact value for caot be writte as a whole umber or a decimal, as it does t repeat or termiate, ad is most simply writte exactly as its radical form:. Example 6 a) Fid two whole values to approximate betwee. b) Use a calculator to approximate the to the earest thousadth.
3 Solutio: a) Sice is betwee 6 ad, is betwee 6 ad, or is betwee ad (but closer to ). O a scietific calculator, the square root key is usually a d fuctio above the x (square) key. For a scietific calculator (based o a TI-0), press: ON,,, d, x (ad maybe ). For a graphig calculator (based o a TI-8/8), you press: ON, d, x,,, Eter. You should see.98 appear. NOTE: Most calculators are accurate to at least more places tha they display. The three more places for this problem are show here:.98. Sice the earest thousadth meas places after the decimal (or decimal places), we look at the place to the right of that, ad otice its a 8, which is greater tha. So roud the thousadths place up ad drop values right of that. To the earest thousadth,.96 Cautio:.96 is ot correct, sice it is a approximatio. Wheever you roud from your calculator, you should use the approximately equals symbol,. So.96 is the correct otatio. The oly exact way to write is, (or by usig ratioal expoets these will be discussed i future courses). Now Do Practice Exercises.. Fid two whole values to approximate 9 betwee.. Approximate 9 to the earest thousadth. Determie if each of the followig is true or false. State a reaso i either case. 6. The umber... The umber 6 8. c More o Ratioal vs. Irratioal Numbers: A ratioal umber q is ay umber that ca be writte as a ratio of itegers (or quotiet or fractio of itegers), where the deomiator caot be 0. I set-builder otatio: a q q, where both a ad b are iteger ad b 0 b (read as the set of umbers q such that q is equal to a divided by b, where both a ad b are iteger ad b is ot equal to zero ). The decimal form of a ratioal umber always either termiates (i.e., ¾ 0. stops with ) or repeats (i.e.,., repeatig the forever). Note: ay ratioal umber ca always be writte as a fractio composed of itegers. You may recall, if it is a termiatig decimal, you put all the digits after the decimal over the digit followed by that may zeros. Example: For 0., put over followed by two 0 s or 00 (ad reduce) If it s a repeatig decimal, put exactly oe full repeat of it s digits over that same umber of 9 s. Example: For , put oe full repeat (09) over two 9 s (ad reduce) oce it s reduced
4 More examples of ratioal umbers writte i fractio ad decimal form:.,, 0., 9, 0 0 Cautio: some ratioal umber decimal repeats caot be see as easily ad caot be see o a calculator at all. For example, the umber has 6 digits i a sigle repeat, but because / is a fractio of itegers, it obeys the defiitio ad is a ratioal umber. Remider, sice ay atural umber, whole umber, or iteger ca be put over, all of these are also i the set of ratioal umbers. A irratioal umber is ay umber that is ot ratioal. It caot be writte as a ratio of itegers ad its decimal represetatio either termiates or repeats. Some examples of irratioal umbers writte i exact ad approximately equal ( ) decimal form:.6..., π.96..., , (This last oe caot be writte i a exact form.) NOTE: To get a accurate decimal form of ay irratioal umber, use a scietific or graphig calculator, as they are ot easily foud by had., ad, whose decimal repeat is log, are both still ratioal umbers, sice they obey that defiitio (they ca be writte as a ratio of itegers). Cautio: umbers like 9, which is exactly equal to NOTE: Most square roots are irratioal. Oly square roots of perfect squares are ratioal. The easiest way to tell if a umber is irratioal is to check that it is ot ratioal. Ask your self these questios: Ca it be writte as some ratio (fractio) of itegers? Or, if it s hard to tell that, check (o a calculator), does the decimal represetatio clearly termiate or repeat? (Agai, these may ot work if the repeat is a log oe.) Now Do Practice Exercises 8 0. Determie if the followig umbers are ratioal or irratioal. Show your reasoig i either case Exercises for the Radicals Packet, Part : a Fid all square roots.. Fid each square root... 9 b Fid two whole values to approximate each square root betwee... Use a calculator to approximate each square root. Roud the square root to the earest thousadth c Determie if each of the followig umbers is ratioal or irratioal
5 BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook PART - th Roots & Simplifyig Radical Expressios (This portio ca be doe ay time after laws of expoets have bee doe. Time i class hr.) a Square Roots, Reviewed ad Expaded. Recall: Whe we raise a umber to the secod power, c, we say it is squared. Whe we wat to fid what umber was squared, we are fidig a square root. The reversal (or iverse) of squarig is fidig a square root. The square roots of egatives: The square roots of egative umbers are ot real umbers. For example, is ot a real umber - because there s o real umber whose square is ; i.e., there is o real umber x such that x (sice the square of ay real umber multiplies a eve umber of egatives, ad is therefore always goig to be positive (or zero)). Square roots of egatives are imagiary umbers ad will be discussed i later courses. Cautio: they are ot udefied, as they do exist ad are used i some very real applicatios, e.g., the field of electroics.) Radical Notatio expaded: The (pricipal) square root of a positive umber a is the positive umber b whose square is a. I symbols, if both a > 0 ad b > 0, a b if b a, Example ad a b if b a, 0 0, Evaluate each of the followig. a. 9 b. a is ot a real umber. c. 8 Solutio: a. 9 because 9. Solutio: b. because c. 8 is ot real because there is o real umber whose square is 8.. Now Do Practice Exercises. Evaluate Hit, recall.0 is 00
6 b th roots. As oted earlier, fidig the square root is the iverse of squarig a umber. Here we ll exted that idea to work with other roots of umbers. For example, The cube root of a umber is the umber we must cube (raise to the third power) to get that umber (The cube root of 8 is sice 8, ad we write 8.) the fourth root of a umber is the umber we must raise to the fourth power to get that umber, etc. The th root of a, writte a, is the umber x where x a. As before, each part of a radical expressio has a special ame. The parts of a radical expressio for ay root, are the idex,, (the root umber) a, For example, the radical sig or radical,, ad the radicad, a. (Recall, the radicad is the etire expressio uder the radical, eve if it s composed of several terms or factors). The cube root of 6 is writte 6 (Idex is ) ad it represets the umber that would be cubed (or raised to the third power) to get 6. So, 6 because 6. The fourth root of 8 is writte 8 (Idex is ) It represets the umber we would raise to the fourth power to get 8. So, 8 because 8. NOTE: Whe the idex is, it is a square root ad it s ot writte: a a Odd roots of egative umbers, (where the idex is odd), will be both real ad egative. (This is true sice a odd power multiplies a odd umber of egatives, ad is therefore always goig to be egative. So a egative real root will exist i these cases.) For example, because ( ) 6. 6 Eve roots of roots of egative umbers, (where the idex is eve), are still ot real umbers. For example, 6 is ot real because there is o real umber whose fourth power is 6; i.e., there is o real umber that ca be raised to the fourth power ad will result i a egative umber. This is because a eve power of ay real umber multiplies a eve umber of egatives, ad is therefore always goig to be positive or zero. 6
7 The followig table shows the most commoly used roots. You fill i the missig values. Example a. Square Roots Cube Roots Fourth Roots Fifth Roots Evaluate each of the followig. 0 c. d. 6 b. Solutio: a. 6 6 because 6 Solutio: b. 0 because 0 Solutio: c. because ( ) - Solutio: c. 0 6 is ot real because there is o real umber whose fourth power is 6. Now Do Practice Exercises 8. Evaluate if possible To coclude this portio, we develop a geeral result eeded i later courses. Let s start by lookig at two examples., ad ( ) sice ( ) Cosider the value of x where x is positive or egative. I where x,. I ( ) where x, ( ). Here ( ) ( ) (the opposite of ). Comparig the results above, we see that x is x if x is positive or zero, ad x is x (the opposite of x) if x itself is egative, (i.e., raisig to a eve power, ad the takig a eve root forces all thigs positive or zero). From your earlier work with absolute values you may remember that x x if x is positive or zero, ad x x (the opposite of x) if x itself is egative. (i.e., absolute value forces all thigs positive or zero). Sice this same sig patter works for ay eve power ad root, we ca summarize the discussio by writig x x for ay eve idex,, ad ay real umber, x.
8 Example a. Evaluate each of the followig. b. ( ) Solutio: a. because Solutio: b. ( ) Now Do Practice Exercises 9 0. Evaluate or because ( ) 6 or ( ) ( ) NOTE: Roots with odd idices do ot require the absolute value, sice a odd power (ad so a odd umber of egative sigs multiplied) is ivolved. For example:, because ad ( ) because ( ) so x x where the idex,, is odd. Combiig the two cocepts i oe statemet, we ca say x x x where the idex is where the idex is eve odd Example a. Evaluate each of the followig. ( ) b. c. ( ) d. ( ) Solutio: a. ( ) because x x Solutio: b. because x x Solutio: c. ( ) because x x Solutio: d. ( ) because x x Now Do Practice Exercises. Evaluate... ( ). ( ) whe the idex,, is eve (it s the uwritte ro a square root here) whe the idex,, is odd ( here) whe the idex,, is odd ( here) whe the idex,, is eve ( here) c Simplifyig Radical Expressios. For most applicatios, we eed to put aswers with radical expressios i simplest form. The word simplest here just meas the followig three coditios have bee met (the actual result may ot look ay simpler tha whe you started). A radical expressio is i simplest form whe. There are o perfect-power factors (greater tha or equal to the idex) iside a radical.. No fractio appears iside a radical.. No radical appears i the deomiator of a fractio. 8
9 As we will be stickig to square roots i this course, this alters the first coditio to read:. There are o perfect-square factors uder a square root. For example: 9 is ot simplest form, sice 9 is a perfect-square factor, ad 8 is ot simplest form, sice 8, ad is a perfect-square factor. But is simplest form, sice has o perfect-square factor (other tha ). We eed to look at two properties to simplify radical expressios. The first of these will be used to simplify expressios with perfect square factors uder square root. (These properties are directly related to the product law ad quotiet law for expoets you studied previously, as you will see i the ext course.) First, look at the followig example This shows that. Here is the geeral law for this fact. Product Property of Radicals For ay o-egative real umbers a, b, ad positive real umber, a b a b I words, the th root of a product is the product of the th roots (ad visa versa). a + b a + b By lettig a ad b, we ca see why. Cautio: This makes the left side: a +b + 9, whereas the right side becomes: a + b + +, ad clearly 9. The first few perfect squares are listed here as a remider to help you recogize them as factors: X X NOTE: a b a b ad a b a b (The operatio is multiply betwee a umber or variable ad a radical.) Example Simplify each of the followig. a. 9 b. 8 c. d. 08 Solutio: a. 9 sice 9 is a perfect square factor itself Solutio: b. 8 Our goal here is to factor radicads as perfect squares ad o-squares. sice 8 by the Product Property of Radicals, (ote: is a perfect square factor) sice (Evaluate the square roots of perfect squares oly.) Solutio: c. is already simplest form sice has o perfect square factors (other tha ) 9
10 Solutio: d sice 08 is divisible by 9 9 ad the is further divisible by. 9 by the Product Property of Radicals, 6 sice 9 ad. (here, 9 ad are both perfect square factors) NOTE: If you did t otice that 9 divides 08, you ca always break the radicad dow to it s prime factors to simplify the radical. If you use this method, you wat to remove factor pairs from uder the radical (sice they produce perfect square factors). Alterate Solutio: d. 08 sice the prime factorizatio of 08 by the Product Property of Radicals sice ad sice x x for eve 6 Now Do Practice Exercises. Simplify This process also works for variable expressios. For this portio, as it will always be give that variables represet o-egative real umbers, the geeral th root of x becomes more simple: For ay o-egative real umber x ad ay iteger >, x x. Example a. Solutio: a. Simplify each of the followig. Assume that all variables represet o-egative real umbers. 6 x b. b c. x x 08a 6 x x sice x m+ x m x (product law of expoets) x x x by the Product Property of Radicals x x x sice x is assumed to be o-egative, x x x Solutio: b. b b b sice b m+ b m b (product law of expoets) b b by the Product Property of Radicals for ay. b b sice ad b is o-egative, b b for ay. b b by commutig. b b by the Product Property of Radicals. 0
11 Solutio: c. 08a Write the radicad as a product of squares times o-squares. 9 a a a sice 08 9 by earlier work, ad a m+ a m a (product law of expoets used repeatedly) 9 a a a by the Product Property of Radicals (split the product up) a a a sice 9, ad, a a whe a is o-egative (simplify the squared parts) a a a by commutig (to put ay radicals at the right ed) 6a a write the repeated factors as a expoet, ad put the ad a uder oe radical (usig the Product Property of Radicals i reverse) Now Do Practice Exercises 8 0. Simplify. Assume that all variables represet o-egative real umbers. 8. 9x 9. 8c 0. m So far we ve oly dealt with the first coditio for simplest form (o square factors iside a square root). Before workig o problems ivolvig the secod ad third coditios, we ll eed to look at aother property. Look at the followig two expressios: 6 00 Thus, (by earlier work) ad 6 6 which gives us the ext geeral rule Quotiet Property of Radicals For ay o-egative real umber a, ad positive real umbers b ad, a a b I words, the th root of a quotiet is the quotiet of the th roots (ad visa versa). This property will be used to simplify radicals ivolvig fractios to meet the last two coditios. A radical expressio is i simplest form whe. No fractio may remai iside a radical.. No radical may remai i the deomiator of a fractio. Example 6 a. 6 Simplify. Assume that all variables represet o-egative real umbers. (Remember, that just meas that x x here, istead of x.) b. 9 c. x
12 Solutio: a. 6 6 We must do somethig, sice we ca t leave a fractio uder a radical. by the Quotiet Property of Radicals We re ot fiished, sice ow we have a radical i the deomiator. sice 6 ad We re still ot doe, sice this reduces. sice NOTE, sice 6 is divisible by, we could have performed that step first, (usig the order of operatios rules). Alterate Solutio: Solutio: b. Solutio: c. 6 sice 6 ad both are iside the radical. sice 9 9 We must do somethig, sice we ca t leave a fractio uder the radical. by the Quotiet Property of Radicals sice 9 x x We re ot fiished, sice ow there s a radical i a deomiator. by the Quotiet Property of Radicals x sice x by the Product Property of Radicals x sice, ad x is o-egative, x x for ay. x by commutig Now Do Practice Exercises. Simplify. Assume that all variables represet o-egative real umbers x 9 You may have oticed the above examples all had perfect squares i the deomiator. If the deomiator is ot a perfect square, we will eed to apply both the Quotiet Property of Radicals ad the Product Property of Radicals i a process called ratioalizig the deomiator. This ivolves multiplyig by the deomiator over itself (still as a radical) to force the deomiator to be a perfect square factor uder the radical. This techique rewrites the fractio as its equivalet with a ratioal umber i the deomiator (ad thus satisfyig coditio leave o radical i the deomiator.)
13 For Example: To simplify the followig expressio,, first use the Quotiet Property of Radicals to rewrite the fractio uder the radical as a quotiet of radicals:. This is ot simplest form, sice there is a radical i the deomiator (see coditio ). To remove it, we eed to multiply by a factor that will make the deomiator a perfect square. Here, multiplyig by sice factor i a fractio over itself (to multiply by the umber, which is ok). would do the trick,. But, we ca t just arbitrarily itroduce a factor of. We eed to use this I geeral, if a 0, the a a a (This follows from the Product Property of Radicals ad the fact that x x whe x is o-egative.) So, to fiish the example, i effect, multiplyig by i the form fractio multiplicatio sice This example ow meets all three coditios for simplest form. Example a. Solutio: a. Solutio: b. Simplify. Assume that all variables represet o-egative real umbers. b. x by the Quotiet Property of Radicals multiplyig by i the form by the Product Property of Radicals ad x x x x by the Quotiet Property of Radicals x x x x multiplyig by i the form by the Product Property of Radicals ad ad fractio multiplicatio for o-egative x ad fractio multiplicatio x x for o-egative x
14 Now Do Practice Exercises 6. Simplify. Assume that all variables represet o-egative real umbers... Exercises for the Radicals Packet, Part 6. y a ad b, ad Part packet: Evaluate if possible a ad b, ad Part packet: Which of the followig roots are ratioal umbers ad which are irratioal umbers? b : Evaluate each of the followig expressios... ( ). ( ) 9. ( ). ( ) a ad b : Fid the two expressios that are equivalet. (Hit: evaluate each expressio if possible first.).. 6,,, 6,..,,,, ,000,, 00, 0,000, ,000 c : Use the Product Property of Radicals to simplify the followig expressios. Assume that all variables represet o-egative real umbers x. 98m a a b c : Use the Quotiet Property of Radicals to simplify the followig expressios
15 c : Use the Quotiet ad Product Properties of Radicals to simplify the followig expressios. Assume that all variables represet o-egative real umbers a.. x Combiig Cocepts.! This symbol meas you eed to write the aswer i complete seteces.!. Explai why x.. x 8s does ot equal x for all real umbers. Give a example to support your reasoig.! 6. Why is the use of absolute value ot required for the expressio x!. Does the th root of x always exist? Why or why ot?! 8. Why will writig 0 0 whe is odd?. ot help i writig the expressio i simplified form? Evaluate each of the followig expressios. Assume that all variables represet ay real umber. ( a + b) 60. ( a + b) ! Decide whether each of the followig is already i simplest form. If ot, explai what eeds to be doe m 6. 8 ab 98x y 6xy x x
16 Begiig Algebra Roots ad Radicals Packet Part - Review of Square Roots & Irratioals Selected Aswers Practice Exercises:. iverse;. 0;. ad ;. radical sig or radical; 9. 8;. 0;. ¾;..8;. Sice , ad 6 asks just for the positive root, 6 8 is a true statemet. 9. Sice is ot a perfect square, must be irratioal. Alterately, 6.80, which is a o-termiatig, o-repeatig decimal, so must be irratioal. Exercises:. ad ;. ;. 6 ad ;..66; 9. (.66, a o-termiatig, o-repeatig decimal, or, is t a perfect square), irratioal; Begiig Algebra Roots ad Radicals Packet Part th Roots & Simplifyig Radical Expressios Selected Aswers Practice Exercises:. ot real. ½.. ot real c c. x.. Exercises:. 0. ot real. ot real. 9. ot real or 9 is ot a perfect square, irratioal or 9 is ot a perfect cube, irratioal 9., ratioal or / is ot a perfect square, irratioal., ratioal i # -, these results are i the order preseted:.,, ot real the first two are equivalet.,, the first two are equivalet. 00, 00, the first two are equivalet 9.. x. a 6a.. 9. a. 6. x 6.! Aswers vary.! Aswers vary 9. a + b 60. a + b! 6. This is already i simplest form.! 6. This is ot simplest sice there s still a perfect square factor (9x ) uder the radical. The simplest form is y. 6