Surds and Indices. Surds and Indices. Curriculum Ready ACMNA: 233,
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1 Surs n Inies Surs n Inies Curriulum Rey ACMNA:, 6
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3 Surs SURDS & & Inies INDICES Inies n surs re very losely relte. A numer uner (squre root sign) is lle sur if the squre root n t e simplifie. Inies (the plurl of inex) is nother wor for exponent. For exmple, the inex of is n is sur. Answer these questions, efore working through the hpter. I use to think: Wht is rtionl numer? Whih one of these is true: # or ? Why re n equl? Answer these questions, fter working through the hpter. But now I think: Wht is rtionl numer? Whih one of these is true: # or ? Why re n equl? Wht o I know now tht I in t know efore? 00% Surs & Inies Mthletis 00% P Lerning K SERIES TOPIC
4 Surs & Inies Bsis All rel numers n e lssifie s either rtionl or irrtionl. Wht is Rtionl Numer? Rtionl numers re ny numers tht n e written s frtion where n re integers n! 0. This mens tht rtionl numers inlue: Oviously ll frtions sine y efinition they re written s where n re integers n! 0. All integers sine ny integer n e written s frtion with enomintor. Any eiml numer tht termintes (eg. 0. ) or reur (eg. 0. o ). 9 If numer n e written s terminting or reurring eiml, then it is rtionl numer. If the numer oes not terminte or repet, then it is not rtionl. These numers re lle irrtionl numers. Are these numers rtionl or irrtionl? This n e written s 0. o o This is reurring eiml ` is rtionl numer ` 0. o o is rtionl numer 0. o o n e written s 99 r 0.6 r This is non-terminting, non-repeting eiml ` r is n irrtionl numer This eiml termintes ` 0.6 is rtionl numer 0.6 n e written s Surs The symol x mens the positive squre root of x. There re three ses for numer uner. If x is perfet squre then the squre root will e rtionl numer. (eg. ) If x is not perfet squre, then the squre root is irrtionl. For exmple is irrtionl. These types of irrtionl numers re lle surs. It s esy to see is irrtionl, sine.6... is nonreurring n non-terminting eiml. If x is negtive, then rel numer. x is unefine euse there is no squre root of negtive numer whih is K 00% Surs & Inies SERIES TOPIC Mthletis 00% P Lerning
5 Surs & Inies Questions Bsis. Why re ALL integers rtionl numers? Give n exmple.. Ientify whether these numers re rtionl or irrtionl. Give reson e 0... f 0.. Ientify whether eh of the following numers is rtionl, irrtionl or unefine (no rel solution). Give reson e f 6 g 0 h Between whih two integers re these surs? % Surs & Inies Mthletis 00% P Lerning K SERIES TOPIC
6 Surs & Inies Knowing More Multiplying Surs Look t this exmple: # 6 # 8 6 ^# 6h So it s esy to see # 6 ^# 6h. This is true for ll numers n the rule for multiplying surs is: # n # ( ) Here re some exmples: # 0 6 # 8 # 6 # 6 # 6 # 0 6# 8# 6# 6 # (Cn't e simplifie further) The multiplition rule n lso e use to simplify surs. If the numer uner the n e written s prout of perfet squre n non-perfet squre then the sur n e simplifie. Here is n exmple: # # Perfet squre # non-perfet squre Simplest sur form Look for multiples of, 9, 6,, 6 et, sine these re perfet squres. Simplifiy these surs: 8 # # 9 # # # 9 # K 00% Surs & Inies SERIES TOPIC Mthletis 00% P Lerning
7 Surs & Inies Knowing More Diviing Surs Look t this exmple: 6 6 So it's esy to see tht 6 6. This is true for ll numers n the rule for iviing surs is: Here re some exmples: 80 ' Wht if there is Numer Outsie the Sur? If there re numers outsie of the sur then they re multiplie or ivie seprtely. Here re some exmples: ( ) # 8 # ' # # 6# 8# # # # 6 96( 6 # 6 ) 96( 6) % Surs & Inies Mthletis 00% P Lerning K SERIES TOPIC
8 Surs & Inies Knowing More Aing n Sutrting Surs Look t these exmples: n So its esy to see tht 9 + is not equl to 9+. This is true for ll numers, so for ny numers n +! + This sign mens "not equl to" The sme n e si for sutrtion. -! - Like Terms Surs n e e n sutrte if they re like terms. Like terms hve the sme numer uner the. Here re some exmples: Simplify the following expressions: Like Terms Like Terms Like Terms - These re unlike terms n nnot e simplifie further Sometimes the surs will hve to e onverte to simplest sur form efore like terms n e foun. Convert the following to simplest sur form n then simplify: Convert to simplest sur form 9# - + # - + Simplify like terms Convert to simplest sur form 6 # + 9# - # + # Simplify like terms 8 - K 6 00% Surs & Inies SERIES TOPIC Mthletis 00% P Lerning
9 Surs & Inies Questions Knowing More. Fin the following prouts. Simplify to n integer if possile: # 8 ^ h 8 # 8 # 6 e # 0 # f # 6 #. Fin the following quotients. Simplify to n integer if possile: e 00 9 f % Surs & Inies Mthletis 00% P Lerning K SERIES TOPIC
10 Surs & Inies Questions Knowing More. Write the following in simplest sur form: e 00 f. Simplify the following into simplest sur form: 0 # 8 # 0 # 0 # e 9 f 6 ' K 8 00% Surs & Inies SERIES TOPIC Mthletis 00% P Lerning
11 Surs & Inies Questions Knowing More. Fin the like terms in eh of the following:,,,, -,,,, -,,,, 6. Simplify these expressions, if possile: , Fin the simplest sur form n then simplify the following s muh s possile: % Surs & Inies Mthletis 00% P Lerning K SERIES TOPIC 9
12 Surs & Inies Using Our Knowlege Prentheses with Surs The istriutive lw sys tht if term multiplies prentheses, then it must multiply ll the terms in the prentheses. For exmple: (x + y) x + y The sme is true if ny of the terms ontin surs. Here re some exmples: Simplify the following: ^ + h ^ + h # + # + # + # 8 + Binomil Prouts with Surs A inomil prout ours when prentheses with two terms re multiplie together. For exmple ( + )(x + y) x + y + x + y The sme is true if ny of the terms ontin surs. Here re some exmples Simplify the following: ^ + h^ - h ^ 6 + h^ 6 + h (simplest sur form) If prentheses re squre, simply multiply the prentheses with itself. For exmple n now the metho in the previous exmples n e use. K ^ + h ^ + h^ + h 0 00% Surs & Inies SERIES TOPIC Mthletis 00% P Lerning
13 Surs & Inies Using Our Knowlege Conjugtes The inomils + n - iffer only in the sign of the seon term. These inomils re lle onjugtes of eh other. When fining onjugtes, rememer tht only the sign of the seon term hnges, not the signs of oth terms. Fin the onjugte of the following: Theonjugte is - Theonjugte is + Theonjugte is - A Shortut to Multiply Binomil with its Conjugte The prout of ( + ) n its onjugte ( - ) n e simplifie like this: Fin the following prouts: ^+ h^- h This is just the ifferene of the squres of the terms in the originl prentheses. Here re some exmples with surs: ^ + h^ - h ^ h -^ h - 8 ^ - h^ + h ^ h -^ h The prout of sur n its onjugte will lwys e the ifferene of the squre of the first term n the squre of the seon term ( + )( - ) ( - ) Sine the squre of sur is equl to the numer in the sur ^ h, the prout of sur n its onjugte is lwys rtionl numer. 00% Surs & Inies Mthletis 00% P Lerning K SERIES TOPIC
14 Surs & Inies Questions Using Our Knowlege. Simplify these prouts to simplest sur form: ^ + h ^ - h ^ + h 0^6-8 h e ^ + h f ^ + - h g x ^ + h h p^ p - 0h i ^ 6 - h j m^m + mh K 00% Surs & Inies SERIES TOPIC Mthletis 00% P Lerning
15 Surs & Inies Questions Using Our Knowlege. Fin these inomil prouts n simplify to simplest sur form: ^ + h^ + 6h ^ - h^ + 0h ^ - 0h^ + h ^ + h^6 - h e ^8 - h^ 0 - h f ^ 6 + h^ - 6 h g ^ + h h ^ - h i ^ + h^ - h j ^ x + h^ x - h 00% Surs & Inies Mthletis 00% P Lerning K SERIES TOPIC
16 Surs & Inies Questions Using Our Knowlege. Write own the onjugte of the following: Fin the prout of the following with its onjugte: e + f + g m - n h + K 00% Surs & Inies SERIES TOPIC Mthletis 00% P Lerning
17 Surs & Inies Thinking More Rtionlising the Denomintor Frtions with rtionl enomintors (like ) re esier to use in lultions thn frtions with irrtionl enomintors (like ). It s importnt to know how to hnge from n irrtionl enomintor to rtionl enomintor. To hnge frtion with n irrtionl enomintor to n equl frtion with rtionl enomintor, multiply the top n ottom of the frtion y the sur in the enomintor. Chnge these frtions into EQUAL frtions with rtionl enomintor Irrtionl enomintor The surin the enomintor is The surin the enomintor is ` multiply top n ottom y ` multiply top n ottom y ^not h # Rtionl enomintor 0 # Multiplying the top n ottom y the sme numer is the sme s multiplying the frtion y. So the originl frtion n the rtionlise frtion re equl. Sine the frtions with irrtionl enomintors hve een hnge to equl frtions with rtionl enomintor, this metho is lle rtionlising the enomintor. Rtionlise eh enomintor n then simplify + 0 Step : Rtionlise the enomintor of eh frtion # + # Step : Fin ommon enomintor n simplify % Surs & Inies Mthletis 00% P Lerning K SERIES TOPIC
18 Surs & Inies Thinking More Wht if the Denomintor Contins n Irrtionl Binomil? Sometimes the enomintor hs inomil ontining surs, like + To rtionlise the enomintor, multiply the top n ottom of the frtion y the enomintor s onjugte. Rtionlise the enomintor of the following: The enomintor's onjugte is + The enomintor's onjugte is 0 - ` multiply top n ottom y + ` multiply top n ottom y # # Rtionlise eh enomintor n then simplify Step : Rtionlise the enomintor of eh frtion - # + + # ^ + h ( + ) + ^ h - ^ h ^ h - ^h Step : Fin ommon enomintor n simplify ^ + 0h K 6 00% Surs & Inies SERIES TOPIC Mthletis 00% P Lerning
19 Surs & Inies Thinking More Surs n e written s Frtionl Inies 'inies' is nother wor for 'exponents' or 'powers' Aoring to the lws of inies `x j # x x Aoring to the lws of surs ^ xh x This mens tht x x If the inex of ny numer is then the squre root is foun. Simplify: Aoring to the lws of inies ^xh # x x Aoring to the lws of surs ^ xh x This mens tht x x If the inex of ny numer is then the ue root is foun. Simplify: So, from the ove it is esy to see: x mens the squre root of x. x mens the ue root of x. Wht out if the inex is or? 00% Surs & Inies Mthletis 00% P Lerning K SERIES TOPIC
20 Surs & Inies Thinking More In similr wy, rule oul e foun for x, x, et. The rule for these inies is: x n n x In ition m n m m x xn n m m n m ` j ^ xh n x n ^x h n ^ x h So the rule for ny numer whih hs ny frtion n m s the inex is: m n n m n m x ^ xh x Here re some exmples Simplify: 6 6 or ^ h - rememer: -n x n x Write in inex form: ^ h Simplify: K 8 00% Surs & Inies SERIES TOPIC Mthletis 00% P Lerning
21 Surs & Inies Questions Thinking More. Rtionlise the enomintors of these frtions n simplify s muh s possile e + f 6 g + h + 8 i m j x x 00% Surs & Inies Mthletis 00% P Lerning K SERIES TOPIC 9
22 Surs & Inies Questions Thinking More. Simplify these s muh s possile, y rtionlising the enomintor of eh frtion Rtionlise the enomintor of the following: K 0 00% Surs & Inies SERIES TOPIC Mthletis 00% P Lerning
23 Surs & Inies Questions Thinking More. Simplify the following s muh s possile, y rtionlising the enomintor of eh frtion % Surs & Inies Mthletis 00% P Lerning K SERIES TOPIC
24 Surs & Inies Questions Thinking More. Write the following in sur form: x ^xh e 00 f ^qh 6. Write the following in inex form: 6 8 e y f p g n h p x q K 00% Surs & Inies SERIES TOPIC Mthletis 00% P Lerning
25 Surs & Inies Questions Thinking More. Fin the vlue of the following (without lultor): 6 9 e f 00 g 0000 h 6 6 # i 8 ` j j ^ # h 00% Surs & Inies Mthletis 00% P Lerning K SERIES TOPIC
26 Surs & Inies Questions Thinking More 8. Show tht is rtionl numer. K 00% Surs & Inies SERIES TOPIC Mthletis 00% P Lerning
27 Surs & Inies Answers Bsis: Bsis:. A rtionl numer is one tht n e written s frtion with integers in the numertor n enomintor. Every interger like - 0 n e written s frtion with in the enomintor, like - 0. This mens tht ll integers re rtionl... g h 0 is rtionl numer is unefine s there is no rel numer tht n e multiplie y itself to give is etween n.. is rtionl numer s it is frtion with integers in the numertor n enomintor. 0 is etween 6 n. is etween 0 n. 0is rtionl s it n e expresse s 0. 9 is etween 9 n is n irrtionl numer s it is non repeting eiml with no en. Knowing More: 0.f is rtionl numer. Although it oes not hve n en, it oes repet. It n e written s. 9. e 0 0 f 0 e f 0. ooo is repeting infinite eiml, so it is rtionl. It n e written s is finite eiml (it hs n en), so it is rtionl. It n e written s e 0 f is unefine s there is no rel numer tht n e multiplie y itself to give - 0. e 0 f 6 is n irrtionl numer. 9 is rtionl numer.. 0 e 00 is rtionl numer. is n irrtionl numer. e f 0 f 6 is rtionl numer. 00% Surs & Inies Mthletis 00% P Lerning K SERIES TOPIC
28 Surs & Inies Answers Knowing More: Using Our Knowlege:.,, Like Terms Like Terms,, -, Like Terms Like Terms Like Terms,,, -, Like Terms,,, e f g h i j x - Like Terms prout prout 68 prout - prout 8 -. e prout 96 - f prout g prout m- 9n Using Our Knowlege: h prout Thinking More: e g i x + x j m m + 6m f h - p- 0p e + f 6 0 K 6 00% Surs & Inies SERIES TOPIC Mthletis 00% P Lerning
29 Surs & Inies Answers Thinking More:. g 6 h i m m j Thinking More: 6 e f g 000 h + i j 6. ^ - h x x e 0, 000, 000, 000 f ( q) 6. e y f p g n h q x p 00% Surs & Inies Mthletis 00% P Lerning K SERIES TOPIC
30 Surs & Inies Notes K 8 00% Surs & Inies SERIES TOPIC Mthletis 00% P Lerning
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