Simplifying Algebra. Simplifying Algebra. Curriculum Ready.

Transcription

1 Simplifying Alger Curriculum Redy

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3 This ooklet is ll out turning complex prolems into something simple. You will e le to do something like this! ( 9- # + 4 ' ) ' ( ) ' ' Give this go! Q Clculte the perimeter of the octopus picture when x. x 8x + 5x x Follow me for gret wy to solve this Mthletics Pssport P Lerning I SERIES TOPIC

4 How does it work? Multiplying nd dividing # & ' Multiplying Dividing # cn e written s ' cn e written s (lpheticl order preferred) '! ' The order doesn t mtter The order does mtter Here re some exmples involving multiply nd dividing. Simplify ech of these (i) 6 # 7x 6 7x 6 7 # # # x Multiply the numers together 4x 6 lots of 7x gives totl of 4x The nottion for squring numers is lso used in lger. Write the letters in lpheticl order. (ii) p # 8 # q # p p # 8 # q # p 8 # p # p # q 8 # # p 8pq q Put the numer first nd vriles in lpheticl order p p p # (sme s for numers) Dividing cn e esier to follow y writing in frction form. (iii) 48x ' 8 Rememer 48 numertor 8 denomintor 48x ' 8 48x x 8 6x 6x Write the division s frction 8 divides evenly into the numertor nd denomintor Simplified frction fter dividing I SERIES TOPIC Mthletics Pssport P Lerning

5 How does it work? Here re some exmples requiring more steps. Fully simplify these expressions Expnding the top nd ottom first cn help you see wht to cncel out when simplifying. (iv) 8xy 7x 8xy 7x 8 x x y 7 # x # # # Expnd the numertor nd denomintor 4 8 x x y 7 # x # # # Simplify the numers (7 divides into oth evenly) 4 x x y x # # # Cncel out ny pirs of mtching vriles 4 x # # y The denomintor ecomes fter cncelling 4xy Simplify y re-writing without the multipliction signs You cn simplify trickier frctions using the sme method. (v) # # # 0 # # # # Expnd the numertor nd denomintor 5 # # # 0 # # # # 4 Simplify the numers (5 divides into oth evenly) # # # 4 # # # # Cncel out pirs of mtching vriles 4 # Wht is left fter cncelling 4 Re-write without the multipliction sign Mthletics Pssport P Lerning I SERIES TOPIC

6 How does it work? Multiplying nd dividing SIMPLIFYING *SIMPLIFYING *SIMPLIFYING * Simplify these products: Multipliction & Division 4 6 # # 7 # # c 5c # d 5d # # e x # 9x # f p # r # q g g g # # g Hint: g h 4m # n # m i 4# ^ ch# ^ h Be creful with negtive signs 4 I SERIES TOPIC Mthletics Pssport P Lerning

7 How does it work? Simplify these: 4 ' Multiplying nd dividing 7 '^ 9h c xy ' 4y d 6 # ^ gh 4 e p 4p f xy x 4 Write these in expnded form nd then simplify: # 5m # 8n n # 0m Mthletics Pssport P Lerning I SERIES TOPIC 5

8 How does it work? Multiplying nd dividing 4 Keep powering on y trying these trickier ones! Write in expnded form nd then simplify: 8 6xy # y 4xy 6 I SERIES TOPIC Mthletics Pssport P Lerning

9 How does it work? Adding nd sutrcting If the vrile prts re exctly the sme, the terms re clled like terms. + & Like terms: x -x y -4y Like terms Like terms Like terms Not like terms: pq pr p -4q Not like terms Simplify these s much s possile. Not like terms Only like terms in n expression cn e dded or sutrcted. Not like terms Mke sure to include the sign in front of ech term when ringing like terms together. (i) 7n+ 6m n The + sign is hidden 7n+ 6m n 7n- n+ 6m Like terms 4n+ 6m This cn't e simplified further. Like terms grouped together Comine the like terms There cn e more thn one pir of like terms. (ii) Like terms Like terms is hidden + The positive term is usully written first to look neter Like terms grouped together Like terms cn hve their vriles written in different order. (iii) 9jk + 4kj jk + 9jk + 4kj jk+ 9jk + 4jk jk+ Both like terms written in lpheticl order Like terms jk jk+ Like terms (iv) xy + xz yx + xz + xy xz Like terms xy + xz yx + xz + xy xz xz + xz xz xz Mthletics Pssport P Lerning I SERIES TOPIC 7

10 SIMPLIFYING *SIMPLIFYING *SIMPLIFYING * How does it work? Adding nd sutrcting Simplify these expressions: + 4d + d Adding nd Sutrcting c 5x+ y+ x 7y d 4m 0 + 4m 8 e 4pq p 7pq+ 9p f 6y y+ 8y + 4y g h w 5w w w Simplify: 4mn m+ nm n 9s 6s+ s 8+ s c xy 5xy + 0xy 9xy d c + 5 c + c + c 8 I SERIES TOPIC Mthletics Pssport P Lerning

11 How does it work? Comining the sic opertions These questions include mix (or comintion) of multiplying, dividing, dding nd sutrcting. Simplify the numertor nd denomintor first, then simplify the frction for ech of these (i) xy 5xy 4x xy 5xy 4x 8xy 4x Like terms in the numertor comined 8 # x # y 4 # x Expnd the numertor nd denomintor 8 4 x y # x # # Simplify the numers where possile x x # # y Cncel out vriles where possile y y (ii) 6mn 8mn + 4mn 6mn 8mn + 4mn 6mn mn Denomintor simplified 6 # m # n # n # m # n Expnd the numertor nd denomintor 6 m n n m n # # # # # Simplify the numers The is hidden m # n # n # m # n Cncel out mtching vriles n n Put the negtive sign out the front By the wy, n cn lso e written s n. Mthletics Pssport P Lerning I SERIES TOPIC 9

12 How does it work? Simplify: Comining the sic opertions 7c+ c 9 5 5d 5d c 8 6 d 4 # 4 Simplify: 4mn + nm 0m x + 8x 8x c x+ 7x x 6x d 4pq 8pq p # 8q 0 I SERIES TOPIC Mthletics Pssport P Lerning

13 How does it work? Comining the sic opertions Try simplifying these slightly trickier ones: 6m + 4m 8m 4 4 q + 8q 9q 6q 9 9 c COMBO TIME * COMBO TIME * COMBO TIME *.../.../0... Mthletics Pssport P Lerning I SERIES TOPIC

14 Where does it work? Multipliction rule for powers Index lws re used for terms tht contin powers (lso clled indices or exponents ). Power m Powers # + # 5 + Bse Simplify these expressions y writing s single term m n m+ n # (i) # 5 Short cut 5 # ^ # # # # h # ^ # # h # # # # # # # ` 5 5+ # 8 If there re numers in front of ech term, multiply them s you would normlly. (ii) p # 6p 4 Short cut 4 p # 6p # ^p # p # p # ph # 6 # ^p # ph S 4 # 6 # p # p # p # p # p # S p # Re-group with numers first 8p 6 ` 4 4 p # 6p ^ # 6h^ p # p h 8p 4+ 8p 6 This rule nd those tht follow in this ook only work when the ses re the sme. I SERIES TOPIC Mthletics Pssport P Lerning

15 Where does it work? A numer written in front of n lgeric term is clled the coefficient of tht term. 7 or 5m The coefficient is 7 The coefficient is 5 In ech of these the se, power nd coefficient is identified: (i) power coefficient se coefficient se power (ii) m 5 m 5 power coefficient se coefficient se m power 5 (iii) 4x 5 4x x power coefficient se coefficient 5 4 se x power (iv) v# v# v # v# v# v# v power coefficient coefficient se se v power Mthletics Pssport P Lerning I SERIES TOPIC

16 Where does it work? Multipliction rule for powers Write down the se, index nd coefficient for ech of these: 4x n m 4 c c 4 d # # 9 se se se se index index index index coefficient coefficient coefficient coefficient Hint: Wht numer does not lwys need to e written? Hint: Write in index form. Write these in index form: h # h # h # h s # s # s c # # # # # d p # p # p q # q e x # x # x # x # # f 5# 5# y # y # y # y 4 I SERIES TOPIC Mthletics Pssport P Lerning

17 Where does it work? Multipliction rule for powers 4 x # x 5 9 # c y # y # y 4 d n # n m # m # m # m 4 Simplify: 6y # y 5 5h 8 # 4h 8 c.5p # 4p d x # x # 5x Mthletics Pssport P Lerning I SERIES TOPIC 5

18 Where does it work? Division rule for powers Dividing is opposite to multiplying. ' When the power is, it is usully hidden. m n m n ' Powers ' ` or 5x 5x Simplify these fully using the division rule (i) ' 7 4 Short cut ' # # # # # # # # # # # # # # # # # # # # 7 in expnded form 4 in expnded form Cncel out pirs of vriles ` ' If there re numers in front of ech term, simplify the frction s you would normlly. (ii) 5 ' k 4k 5 k ' 4k k 4k 5 Short cut k k k k k 4 # k # # # # # k k k k k 4 # k # # # # # k k k k # # # # Expnded form Simplify y cncelling out k 4 ` k ' 4k ^' 4hk 5 5 k 4 6 I SERIES TOPIC Mthletics Pssport P Lerning

19 Where does it work? Simplify: Division rule for powers w # w # w # w # w w # w # w 7 d d '.../.../ c m ' m d p p 6 4 Simplify: 6 ' v 4 8v c 70m ' 0m 4 4 d d 8d 6 Mthletics Pssport P Lerning I SERIES TOPIC 7

20 COMBO TIME * COMBO TIME * COMBO TIME * Where does it work? Comining multipliction nd division rules Simplify ech of these: ^m m # m # mh m # ' y y y 8 5 # 9.../.../ c ' # d w 4 # w w # w # w # w Simplify: k 9k k 4 # 0n ' n n I SERIES TOPIC Mthletics Pssport P Lerning

21 Where does it work? Comining multipliction nd division rules Keep the momentum going y simplifying these: 5 # 7 8 7x # 8x 8x ' 4x 8 4 c 4m # 6n n # m 5 4 Mthletics Pssport P Lerning I SERIES TOPIC 9

22 Where does it work? Indices cogs puzzle To mke the cogs rotte, missing simplifictions must e completed correctly. Fill in the missing simplifictions 5 x y # 5x y # 5# x # y # y # x # y 6 7 5x y 4 # # # 87 n m n m # + # # x # x # x 4 # m n m n # # # # + + # mn ' 8mn 6 6 ' x y ' 8xy ^ ' h' 5 5 ^56 ' 8hx y # ' 5 - # # 4 7xy 9 6 ' 6 n m n m ' 5 ' 5 6x 8 4 ' x x m m 0 # m ' m n ' m n 64 0 I SERIES TOPIC Mthletics Pssport P Lerning

23 Where does it work? Power rule for powers # 6 # ^ h ^ h m n m n mn Power # Everything inside the rcket is ffected y the power outside: # # ^ h # # ^h x y n x n y n nx ny Simplify these using the power rule (i) ^ h ^ h # # Short cut + + Simplify using the multipliction rule 6 ` ^ h # 6 Every numer or vrile inside the rckets is ffected y the power outside of the rcket: (ii) n 4 ^ h ^ n h n # n # n # n 4 ^ # n h# ^ # n h# ^ # n h# ^ # n h Expnd ech term Short cut n n n n # # # # # # # Re-group n # Simplify 4096n 8 ` ^ n h n 4 # 4 # 4 n n 8 Mthletics Pssport P Lerning I SERIES TOPIC

24 Where does it work? Power rule for powers Simplify: ^ h ^ h j 5 c ^r h d x 4 ^ h Simplify nd clculte the vlue of these: (this is lso clled finding the sic numerl ) ^ h ^ h Simplify: ^ h 5r 4 ^ h # # c ^ h d 4xyz ^ h k 6 I SERIES TOPIC Mthletics Pssport P Lerning

25 Where does it work? Zero rule for powers Simplify 0 using the zero power rule Why? Awesome question! 0 Let's look closely t how we cn nswer this. If the numertor nd denomintor re exctly the sme in frction, it hs vlue of. m ` m Becuse numertor nd denomintor re the sme Let's see wht hppens when we look t the sme frction nd simplify using the division rule. ` If m m m m m ' m- m m 0 nd m m 0 From the division rule Rememer numertor denomintor ` then 0 Simplify these using the zero power rule The position of the power rule is importnt. Rememer, everything inside pir of rckets is ffected y the power outside of the rckets. (i) ( ) 0 0 # 0 # 0 ^h Everything is ffected y the power of # ` 0 ^h (ii) 0 (sme s the lst question without the rckets) 0 0 # 0 Only the vrile is ffected y the power of 0 Mthletics Pssport P Lerning I SERIES TOPIC

26 ZERO RULE FOR POWERS * Where does it work? Zero rule for powers Simplify: x /.../0... c ^5ch 0 d 6m 0 Simplify: 0 9 # 0 0 c 0 6y ^4y h 0 d k ' 7^k h 0 0 e 7 # ^mnh I SERIES TOPIC Mthletics Pssport P Lerning

27 COMBO TIME * COMBO TIME * COMBO TIME * Where does it work? Comining ll the power rules Simplify: g #^g h g 4.../.../ cd 6 5cd ^ h ^ h ' ^xy zh # ^x yzh This one is worthy of n wesome stmp! * AWESOME *.../.../0... * AWESOME * Mthletics Pssport P Lerning I SERIES TOPIC 5

28 Where does it work? Alger terminology puzzle Unscrmle the words elow nd use the letters in the octgons together with the clue to find the nswer. csieind glicer fysilimp nextpone woper elik smert viler se pndexed morf Clue Working together we were le to put this numer out the front with no time wsted! Numer Vrile Answer 6 I SERIES TOPIC Mthletics Pssport P Lerning

29 Wht else cn you do? Perimeter nd re prolems Are Perimeter nd re prolems cn e simplified using your new lger skills. Collecting like terms cn simplify perimeter expressions. Simplify these prolems then clculte using sustitution (i) ) Write simplified version of the perimeter for the tringle elow: x + x 5 4x + 9 Perimeter x+ + x- 5+ 4x + 9 x+ x+ 4x x + 6 Group the like terms ) Find the perimeter if x units: If x, perimeter 8 # + 6 units Sustitute in for x in the perimeter expression 0 units The sic index lws re useful for simplifying re clcultions s shown in the next prolem. (ii) ) Write simple expression for the re of the rectngle elow: 4 Are width # length units 4 # units 4 # # + units units Simplify using the multipliction rule ) Find the re if : If, re # units Sustitute in for in the re expression 96 units Mthletics Pssport P Lerning I SERIES TOPIC 7

30 Wht else cn you do? Perimeter nd re prolems (i) Write simplified expressions for the perimeter of the tringle: 6x + 5 x + (ii) Find the perimeter if x units (i) Write simplified expressions for the perimeter of the rectngle: + 5 (ii) Find the perimeter if 5 units AREA & AREA & PERIMETER.../.../0... PERIMETER 8 I SERIES TOPIC Mthletics Pssport P Lerning

31 Wht else cn you do? Perimeter nd re prolems (i) Write simplified expressions for the perimeter of the rhomus: m + n - (ii) Find the perimeter if m 4 unit nd n units 4 (i) Write simplified expressions for the perimeter of the pentgon: x - y + (ii) Find the perimeter if x 5 nd y 8 units Mthletics Pssport P Lerning I SERIES TOPIC 9

32 Wht else cn you do? Perimeter nd re prolems 5 (i) Write simplified expressions for the re of ech shpe elow: (ii) Clculte the re of ech shpe when 4 units 4 0 I SERIES TOPIC Mthletics Pssport P Lerning

33 Wht else cn you do? Perimeter nd re prolems 6 (i) Write simplified expression for the perimeter of the octopus picture: + x 8x Rememer me? 5x x (ii) Clculte the perimeter of the octopus picture when x * AWESOME *.../.../0... * AWESOME * (iii) For n wesome stmp, write simplified expression for the perimeter of the octopus picture if the length ws hlved nd the width ws douled. Mthletics Pssport P Lerning I SERIES TOPIC

34 Chet sheet Here is summry of the importnt things to rememer when simplifying lger. Multiplying Dividing # cn e written s ' cn e written s (lpheticl order preferred) '! ' The order doesn t mtter The order does mtter Adding nd sutrcting If the vrile prts re exctly the sme, the terms re clled like terms. Like terms: x -x y -4y Like terms Like terms Like terms Not like terms: pq pr p -4q Not like terms Not like terms Not like terms Only like terms in n expression cn e dded or sutrcted. Multipliction rule for powers # m n m+ n # + Division rule for powers Dividing is opposite to multiplying. ' m n m n ' Power rule for powers ^ h 6 # Everything inside the rcket is ffected y the power outside: # ^ h m n m n mn 4 # 4 # 6 8 # # ^ h ^h x y n x n y n nx ny Zero rule for powers ^h 0 Power of When the power is, it is usully hidden: ` or 5x 5x Coefficient The numer written in front of n lgeric term: For the term 7, the coefficient is 7. I SERIES TOPIC Mthletics Pssport P Lerning

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