Chapter. Radicals (Surds) Contents: A Radicals on a number line. B Operations with radicals C Expansions with radicals D Division by radicals

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1 Chter 4 Rdils (Surds) Contents: A Rdils on numer line B Oertions with rdils C Exnsions with rdils D Division y rdils

2 88 RADICALS (SURDS) (Chter 4) INTRODUCTION In revious yers we used the Theorem of Pythgors to find the length of the third side of tringle. Our nswers often involved rdils suh s,,, nd so on. ~` A rdil is numer tht is written using the rdil sign. Rdils suh s 4 nd 9 re rtionl sine 4= nd 9=. Rdils suh s, nd re irrtionl. They hve deiml exnsions whih neither terminte nor reur. Irrtionl rdils re lso known s surds. RESEARCH ² Where did the nmes rdil nd surd ome from? ² Why do we use the word irrtionl to desrie some numers? ² Before we hd lultors nd omuters, finding deiml reresenttions for numers like to four or five deiml les ws uite diffiult nd time onsuming. Imgine hving to find :44 orret to five deiml les using long division! A method ws devised to do this lultion uikly. Wht ws the roess? SQUARE ROOTS The sure root of or is the ositive numer whih oeys the rule =. For to hve mening we reuire to e non-negtive, i.e., > 0. For exmle, = or ( ) =. Note tht A 4=, not, sine the sure root of numer nnot e negtive. RADICALS ON A NUMBER LINE If we onvert rdil suh s to deiml we n find its roximte osition on numer line. ¼ : 0, so is lose to 4. ~` 0

3 RADICALS (SURDS) (Chter 4) 89 We n lso onstrut the osition of on numer line using ruler nd omss. Sine + =( ), we n use right-ngled tringle with sides of length, nd. Ste : Drw numer line nd mrk the numers 0,,, nd on it, m rt. Ste : Ste : Ste 4: EXERCISE 4A With omss oint on, drw n r ove. Do the sme with omss oint on using the sme rdius. Drw the erendiulr t through the intersetion of these rs, nd mrk off m. Cll this oint A. Comlete the right ngled tringle. Its sides re, nd m. With entre O nd rdius OA, drw n r through A to meet the numer line. It meets the numer line t. Notie tht +4 ==( ). Lote on numer line using n urte onstrution. The sum of the sures of whih two ositive integers is? Aurtely onstrut the osition of on numer line. 0 Cn we onstrut the ext osition of on numer line using the method ove? 4 nnot e written s the sum of two sures so the ove method nnot e used for loting 4 ~` on the numer line. However, 4 =, so 4 = +( ). We n thus onstrut right ngled tringle with sides of length 4, nd. Use suh tringle to urtely lote on numer line. ~` DEMO B OPERATIONS WITH RADICALS ADDING AND SUBTRACTING RADICALS We n dd nd sutrt like rdils in the sme wy s we do like terms in lger. For exmle: ² just s + =, + = ² just s 4 =, 4 =. Exmle Simlify: 4 ( ) 4 = = ( ) = + =

4 90 RADICALS (SURDS) (Chter 4) SIMPLIFYING PRODUCTS We hve estlished in revious yers tht: =( ) = = r = Exmle Simlify: ( ) ( ) µ 4 ( ) = = ( ) = = µ 4 = 4 ( ) = =8 Exmle Simlifying: ( ) ( ) ( ) = =9 =8 ( ) = = = 8 Exmle 4 Write in simlest form: 4 With rtie you should not need the middle stes. = = 0 4 = 4 = =

5 RADICALS (SURDS) (Chter 4) 9 Exmle Simlify: = = = = = = 4 = EXERCISE 4B. Simlify: + 8 d + e ( ) f ( + ) g + h + i ( ) j ( + ) + ( ) k ( ) ( ) l ( ) ( ) Simlify: ( ) ( ) ( ) d e ( ) f ( ) g i ( ) j ( ) 4 k Simlify: µ h µ l ( ) (4 ) ( ) d ( ) e ( ) f ( ) g ( ) h ( 0) i ( 0) µ µ µ 0 j 4 k l m ( 4 ) n ( ) o ( ) ( )( ) ( ) r ( ) 4 Simlify: d 9

6 9 RADICALS (SURDS) (Chter 4) Simlify: d e f ( ) g h i j ( ) ( ) k ( ) l ( ) Simlify: 8 8 d 0 e f g 8 h 0 i j 4 k 4 4 l 98 Is 9+ = 9+? Is =? Are + = + nd = ossile lws for rdil numers? 8 Prove tht = for ll ositive numers nd. Hint: Consider ( ) nd ( ). r Prove tht = for > 0 nd >0. SIMPLEST RADICAL FORM A rdil is in simlest form when the numer under the rdil sign is the smllest ossile integer. Exmle Write 8 in simlest form. 8 = 4 = 4 = We look for the lrgest erfet sure tht n e tken out s ftor of this numer. = 4 8= 8 is not in simlest form s 8 n e further simlified into. In simlest form, = 4.

7 RADICALS (SURDS) (Chter 4) 9 Exmle It my e useful Write to do rime 4 in simlest 4 rdil form. = ftoristion of the 4 numer under the = 4 rdil sign. =4 = EXERCISE 4B. Write in the form k where k is n integer: 8 0 d 98 e f 00 g h Write in the form k where k is n integer: 48 d 00 Write in the form k where k Z : d 00 4 Write in simlest rdil form: d e 48 f g 4 h i j 0 k l 000 Write in simlest rdil form + n where, Q, n Z : 4+ 8 e + f g d 8 4 h 00 0 C EXPANSIONS WITH RADICALS The rules for exnding rdil exressions ontining rkets re identil to those for ordinry lger. ( + ) = + ( + )( + d) = + d + + d ( + ) = + + ( + )( ) =

8 94 RADICALS (SURDS) (Chter 4) Exmle 8 Simlify: ( + ) ( ) ( + ) = + ( ) =4+ = + = 4 With rtie you should not need the middle stes. Exmle 9 Exnd nd simlify: ( + ) ( ) ( + ) = + = ( ) = + = + EXERCISE 4C Exnd nd simlify: 4( + ) ( + ) (4 ) d ( 4) e ( + ) f ( ) g ( + ) h ( ) i ( ) j ( ) k ( + ) l ( + + ) Exnd nd simlify: (4 + ) ( ) ( ) d ( + ) e ( ) f ( + ) g ( ) h ( + ) i ( ) j ( +4) k ( ) l ( ) (4 ) Exmle 0 Exnd nd simlify: ( + )( + ) ( + )( ) ( + )( + ) = ( + )+( + ) =+ + + =8+ ( + )( ) =(+ )( + ) = ( + )+( + )( ) =+ =

9 RADICALS (SURDS) (Chter 4) 9 Exnd nd simlify: ( + )( + ) ( + )( + ) ( + )( ) d (4 )( + ) e ( + )( ) f ( )( + ) g ( + )( ) h ( + )( ) i ( + )( +) j ( )( + ) Exmle Exnd nd simlify: ( +) ( ) ( +) =( ) + () + =+ +9 =+ ( ) =( + ) =( ) + ( ) + ( ) = + =8 4 Exnd nd simlify: ( + ) ( +) ( ) d ( + ) e ( ) f (4 ) g ( + ) h ( ) i ( ) j ( +) k ( ) l ( ) Exmle Exnd nd simlify: (4 + )(4 ) ( + )( ) (4 + )(4 ) =4 ( ) = =4 ( + )( ) =( ) =8 9 = Exnd nd simlify: ( + )( ) ( )( +) ( + )( ) d ( 4)( +4) e ( )( +) f ( + )( ) g ( )( + ) h ( + )( )

10 9 RADICALS (SURDS) (Chter 4) i ( + )( ) j ( )( + ) k ( )( + ) l ( + )( ) D DIVISION BY RADICALS In numers like nd 9 we hve divided y rdil. It is ustomry to simlify these numers y rewriting them without the rdil in the denomintor. INVESTIGATION DIVISION BY In this investigtion we onsider frtions of the form where nd re rel numers. To remove the rdil from the denomintor, there re two methods we ould use: ² slitting the numertor ² rtionlising the denomintor Wht to do: Consider the frtion. Sine is ftor of, slit the into. Simlify. Cn the method of slitting the numertor e used to simlify? Consider the frtion. If we multily this frtion y, re we hnging its vlue? Simlify y multilying oth its numertor nd denomintor y. 4 The method in is lled rtionlising the denomintor. Will this method work for ll frtions of the form where nd re rel? From the Investigtion ove, you should hve found tht for ny frtion of the form, we n remove the rdil from the denomintor y multilying y. Sine =, we do not hnge the vlue of the frtion.

11 RADICALS (SURDS) (Chter 4) 9 Exmle Write with n integer denomintor: Multilying the originl numer y or does not hnge its vlue. = = = = = = EXERCISE 4D. Write with integer denomintor: f k g l RADICAL CONJUGATES h 9 d i m n 00 r s e j o t 4 ( ) Rdil exressions suh s + nd whih re identil exet for oosing signs in the middle, re lled rdil onjugtes. The rdil onjugte of + is. INVESTIGATION RADICAL CONJUGATES Frtions of the form + n lso e simlified to remove the rdil from the denomintor. To do this we use rdil onjugtes. Wht to do: Exnd nd simlify: ( + )( ) ( )( +) Wht do you notie out your results in?

12 98 RADICALS (SURDS) (Chter 4) Show tht ( + for ny integers )( nd, the following roduts re integers: ) ( )( + ) 4 Coy nd omlete: To remove the rdils from the denomintor of frtion, we n multily the denomintor y its... Wht must we do to the numertor of the frtion to ensure we do not hnge its vlue? From the Investigtion ove, we should hve found tht: to remove the rdils from the denomintor of frtion, we multily oth the numertor nd the denomintor y the rdil onjugte of the denomintor. Exmle 4 Write 4 with n integer denomintor. µ Ã 4 4 = +! + = 4 9 ( + ) = ( + ) =+ Exmle Write + : with integer denomintor in the form + where, Q. + µ Ã = +! = = = So, = nd =.

13 RADICALS (SURDS) (Chter 4) 99 EXERCISE 4D. Write with integer denomintor: + + d e + f 4 g h + Write in the form + where, Q : 4 + d + Write in the form + where, Q : 4 + d If, nd re integers, show tht ( + )( ) is n integer. Write with n integer denomintor: i + ii. If nd re integers, show tht ( + )( ) is lso n integer. Write with n integer denomintor: i ii + LINKS lik here HOW A CALCULATOR CALCULATES RATIONAL NUMBERS Ares of intertion: Humn ingenuity REVIEW SET 4A Simlify: ( ) µ 4 Coy nd omlete: + =(::::::) Simlify: d 4 Use to urtely onstrut the osition of 0 on numer line using ruler nd omss. d 0

14 00 RADICALS (SURDS) (Chter 4) 4 Write 8 in simlest rdil form. Hene, simlify 8. Write 98 in simlest rdil form. Exnd nd simlify: ( +) ( ) ( + ) d ( ) e ( + )( ) f ( + )( ) Write with n integer denomintor: Write in the form + where, Q : + REVIEW SET 4B Simlify: 8 ( ) d 4 9 Find the ext osition of on numer line using ruler nd omss onstrution. Exlin your method. Hint: Look for two ositive integers nd suh tht =. Simlify: 4 4 Simlify: Write in simlest rdil form: Exnd nd simlify: ( ) ( ) ( ) d ( + ) e ( )( + ) f ( + )( ) Write with integer denomintor: Write in the form + where, Q :

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