SIMPLIFYING ALGEBRA PASSPORT.

Transcription

1 SIMPLIFYING ALGEBRA PASSPORT

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3 This booklet is ll bout turning complex problems into something simple. You will be ble 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 # b cn be written s b ' b cn be written s b b b (lphbeticl order preferred) ' b! b' The order doesn t mtter The order does mtter Simplify 6 # 7x 6 7x 6 7 # # # x Multiply the numbers together 4x 6 lots of 7x gives totl of 4x The nottion for squring numbers is lso used in lgebr. Write the letters in lphbeticl order. Simplify p # 8 # q # p p # 8 # q # p 8 # p # p # q 8 # # p 8pq q Put the number first nd vribles in lphbeticl order p p p # (sme s for numbers) Dividing cn be esier to follow by writing in frction form. Simplify 48x ' 8 Remember 48 numertor 8 denomintor 48x ' 8 48x x 8 6x Write the division s frction 8 divides evenly into the numertor nd denomintor Simplified frction fter dividing 6x I SERIES TOPIC Mthletics Pssport P Lerning

5 How does it work? Expnding the top nd bottom first cn help you see wht to cncel out when simplifying. Simplify 8xy 7x 8xy 7x 8 x x y 7 # x # # # Expnd the numertor nd denomintor 4 8 x x y 7 # x # # # Simplify the numbers (7 divides into both evenly) 4 x x y x # # # Cncel out ny pirs of mtching vribles 4 x # # y The denomintor becomes fter cncelling 4xy Simplify by re-writing without the multipliction signs You cn simplify trickier frctions using the sme method. Simplify 5b 0b 5b 0b 5 # # b # b 0 # # # b # b Expnd the numertor nd denomintor 5 # # b # b 0 # # # b # b 4 Simplify the numbers (5 divides into both evenly) # # b # b 4 # # # b # b Cncel out mtching vribles 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 # b 7 # # b c 5c # d 5d # # e x # 9x # f p # r # q g g g # # Hint: g g h 4m # n # m i 4b # ^ ch # ^ h Be crefulwithnegtive signs 4 I SERIES TOPIC Mthletics Pssport P Lerning

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

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

9 How does it work? Adding nd subtrcting If the vrible prts re exctly the sme, the terms re clled like terms. + & Like terms: x -x b b 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 be dded or subtrcted. Mke sure to include the sign in front of ech term when bringing like terms together. Simplify 7n+ 6m n s much s possible. 7n+ 6m n 7n- n+ 6m Like terms The + sign is hidden 4n+ 6m Like terms grouped together Combine the like terms This cn't be simplified further There cn be more thn one pir of like terms. Simplify + 8b 4+ b s much s possible. Like terms + 8b 4+ b b+ b Like terms is hidden + b b Like terms grouped together The positive term is usully written first to look neter Like terms cn hve their vribles written in different order. Simplify 9jk + 4kj jk+ s much s possible. 9jk + 4kj jk+ 9jk + 4jk jk+ Like terms jk jk+ Both like terms written in lphbeticl order Mthletics Pssport P Lerning I SERIES TOPIC 7

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

11 How does it work? Combining the bsic opertions These questions include mix (or combintion) of multiplying, dividing, dding nd subtrcting. Simplify the numertor nd denomintor first, then simplify the frction. Simplify xy 5xy 4x xy 5xy 4x 8xy 4x Like terms in the numertor combined 8 # x # y 4 # x Expnd the numertor nd denomintor 8 4 x y # x # # Simplify the numbers where possible x x # # y Cncel out vribles where possible y y Simplify 6mn 8mn + 4mn 6mn 8mn + 4mn The is hidden 6mn mn 6 # m # n # n # m # n 6 m n n m n # # # # # m # n # n # m # n Denomintor simplified Expnd the numertor nd denomintor Simplify the numbers Cncel out mtching vribles n n Put the negtive sign out the front Mthletics Pssport P Lerning I SERIES TOPIC 9

12 How does it work? Simplify: Combining the bsic opertions 7c+ c 9 b 5 5d 5d c 8 6 d 4b # 4b Simplify: 4mn + nm 0m b 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? Combining the bsic opertions Try simplifying these slightly trickier ones: 6m + 4m 8m 4 4 b q + 8q 9q 6q 9 9 c b+ 5b 4b 4b 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 m n m+ n # + Simplify # 5 Short cut 5 # ^ # # # # h # ^ # # h # # # # # # # ` 5 5+ # 8 If there re numbers in front of ech term, multiply them s you would normlly. Simplify 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 # p 6 Re-group with numbers first ` 4 4 p # 6p ^ # 6h^ p # p h 8p 4+ 8p 6 This rule nd those tht follow in this book only work when the bses re the sme. I SERIES TOPIC Mthletics Pssport P Lerning

15 Where does it work? A number written in front of n lgebric term is clled the coefficient of tht term. 7 or 5m The coefficient is 7 The coefficient is 5 In ech of these the bse, power nd coefficient is identified: power coefficient bse coefficient bse power m 5 power m 5 coefficient bse coefficient bse m power 5 4x 5 4x x power coefficient bse coefficient 5 4 bse x power Mthletics Pssport P Lerning I SERIES TOPIC

16 Where does it work? Multipliction rule for powers Write down the bse, index nd coefficient for ech of these: 4x n b m 4 c b c 4 d # # 9 bse bse bse bse index index index index coefficient coefficient coefficient coefficient Hint: Wht number does not lwys need to be written? Hint: Write in index form. Write these in index form: h # h # h # h b 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 Simplify ech of these: 4 x # x 5 9 b b # b c y # y # y 4 d n # n m # m # m # m 4 Simplify: 6y # y 5 b 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 ' 7 4 Short cut ' # # # # # # # # # # # # # # # # # # # # 7 in expnded form 4 in expnded form Cncel out pirs of vribles ` ' If there re numbers in front of ech term, simplify the frction s you would normlly. Simplify k 5 ' 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 by 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 b d d '.../.../0... * DIVISION RULE FOR POWERS 4 c m ' m d p p 6 4 Simplify: 6 ' 8 6 b 48v 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? Combining multipliction nd division rules Simplify ech of these: ^m m # m # mh m # ' b y y y 8 5 # 9.../.../ c ' # d w 4 # w w # w # w # w Simplify: k 9k k 4 # b 0n ' n n I SERIES TOPIC Mthletics Pssport P Lerning

21 Where does it work? Combining multipliction nd division rules Keep the momentum going by simplifying these: 5b # b b 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 discs Indices Guy remembers his bsic index lws with the discs shown below. Unfortuntely, some exmples hve been ersed. Help Indices Guy restore his reference discs by using the index rules t the centre of the disc to fill in the gps. 4 4b # 9b # # + 5 xy # 5xy # 5 # x # y # y 5 5 # + x # + y 6 7 5xy 4 # # 87 n m n m # + 4 # mn mn 5 8 x # x # x b # b # # b # b + # + b 5 b 9 6 4mn ' 8mn 6 6 ' x y ' 8xy ^56 ' 8hx 4 7xy 9 6 ' y n m n m ' 5 5 ^ ' h' 5 b b ' b 5 b ' mn ' mn 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 brcket is ffected by the power outside: 4 # 4 # 6 8 # # ^ h ^bh b b x y n x n y n nx ny Simplify ^ h ^ h # # Short cut Simplify using the multipliction rule ` ^ h # 6 Every number or vrible inside the brckets is ffected by the power outside of the brcket: Simplify 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 4 n # 4 # 4 n n 8 Mthletics Pssport P Lerning I SERIES TOPIC

24 POWER RULE FOR POWERS * Where does it work? Power rule for powers Simplify: 5 ^j h b ^b h.../.../0... c 4 ^r h 0.5 d x 4 ^ h Simplify nd clculte the vlue of these: (this is lso clled finding the bsic numerl ) ^ h b ^ h Simplify: ^ # # h b 4 ^5r h c 6 ^ k h d ^4xyzh I SERIES TOPIC Mthletics Pssport P Lerning

25 Where does it work? Zero rule for powers Simplify 0 0 Why? Awesome question! 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 ` Becuse numertor nd denomintor re the sme m Let's see wht hppens when we look t the sme frction nd simplify using the division rule. ` ` If m m m m then 0 m m ' m m 0 m nd m 0 From the division rule Remember numertor denomintor Remember, everything inside pir of brckets is ffected by the power outside of the brckets. Simplify ( ) 0 0 ^h # 0 ` ^h # 0 # Everything is ffected by the power of 0 Simplify 0 (sme s the lst question without the brckets) # Only the vrible is ffected by 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 b /.../0... c ^5ch 0 d 6m 0 Simplify: b 0 b 9 # b 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? Combining 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? Puzzle Time Unscrmble the words below nd use the letters in the octgons together with the clue to find the nswer. csieind glbicer fysilimp nextpone woper elik smert vibler sbe pndexed morf Clue Working together we were ble to put this number out the front with no time wsted! Number Vrible Answer 6 I SERIES TOPIC Mthletics Pssport P Lerning

29 Wht else cn you do? Perimeter nd re problems Are Perimeter nd re problems cn be simplified using your new lgebr skills. Collecting like terms cn simplify perimeter expressions. Check out this problem to see how: (i) Write simplified version of the perimeter for the tringle below: x + x 5 4x + 9 Perimeter x+ + x- 5+ 4x + 9 x+ x+ 4x Group the like terms 8x + 6 (ii) Find the perimeter if x units: If x, perimeter 8 # + 6 units Substitute in for x in the perimeter expression 0 units The bsic index lws re useful for simplifying re clcultions s shown in the next problem. (i) Write simple expression for the re of the rectngle below: 4 Are width # length units 4 # units 4 # # + units Simplify using the multipliction rule units (ii) Find the re if If, re # units Substitute 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 problems (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: b b + 5 (ii) Find the perimeter if b 5 units AREA & AREA & PERIMETER.../.../0... PERIMETER 8 I SERIES TOPIC Mthletics Pssport P Lerning

31 Wht else cn you do? Perimeter nd re problems (i) Write simplified expressions for the perimeter of the rhombus: 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 problems 5 (i) Write simplified expressions for the re of ech shpe below: (ii) Clculte the re of ech shpe when 4 units b 4 0 I SERIES TOPIC Mthletics Pssport P Lerning

33 Wht else cn you do? Perimeter nd re problems 6 (i) Write simplified expression for the perimeter of the octopus picture: + x 8x Remember 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 doubled. Mthletics Pssport P Lerning I SERIES TOPIC

34 Chet sheet Here is summry of the importnt things to remember when simplifying lgebr. Multiplying Dividing # b cn be written s b ' b cn be written s b b b (lphbeticl order preferred) ' b! b' The order doesn t mtter The order does mtter Adding nd subtrcting If the vrible prts re exctly the sme, the terms re clled like terms. Like terms: x -x b b 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 be dded or subtrcted. 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 brcket is ffected by the power outside: # ^ h m n m n mn 4 # 4 # 6 8 # # ^ h ^bh b b x y n x n y n nx ny Zero rule for powers 0 0 Power of When the power is, it is usully hidden: ` or 5x 5x Coefficient The number written in front of n lgebric term: For the term 7, the coefficient is 7. I SERIES TOPIC Mthletics Pssport P Lerning

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36 POWER RULE FOR POWERS * SIMPLIFYING *SIMPLIFYING *SIMPLIFYING * Adding nd Subtrcting.../.../ /.../0... * DIVISION RULE FOR POWERS