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Student Book - Series K-1

Algebraic Techniques

Mathletics Instant

Workbooks

2(a + b)3

(x + 2)(3x + y)

Algebraic techniquesStudent Book - Series K

Contents

Topics

Author of The Topics and Topic Tests: AS Kalra

Practice Tests

Topic 1 - Topic test A __/__/__ Topic 2 - Topic test B __/__/__

Algebraic techniquesMathletics Instant Workbooks – Series K Copyright © 3P Learning

Date completed

Topic 1 - Algebraic expressions __/__/__

Topic 2 - Addition and subtraction in algebra __/__/__

Topic 3 - Multiplication and division in algebra __/__/__

Topic 4 - Index notation and algebra __/__/__

Topic 5 - Revising and combining index laws __/__/__

Topic 6 - Negative indices __/__/__

Topic 7 - Fractional indices __/__/__

Topic 8 - Grouping symbols in algebra __/__/__

Topic 9 - Expanding and simplifying algebraic expressions __/__/__

Topic 10 - Substitution __/__/__

Topic 11 - Addition and subtraction of algebraic fractions __/__/__

Topic 12 - Multiplication and division of algebraic fractions __/__/__

Topic 13 - Factorisation using common factors __/__/__

Algebraic techniques

Algebraic techniquesMathletics Instant Workbooks – Series K Copyright © 3P Learning 1

Topic 1: Algebraic expressions

QUESTION 1 Write algebraic expressions for the following.a The sum of a and b = _____________________ b The product of x and y = _________________c The square of m = ________________________ d The square root of p = _________________e The sum of 7x and 2y = ___________________ f The cube of k = _________________g The square of 5x = _______________________ h The difference between 8p and 3q = ________i The number 3x divided by 7 = _____________ j Nine times the square of a = _______________k Half the number minus 6 = ________________ l Square root of 7, times the number = ________

QUESTION 2 Write the algebraic expressions for the following.a The cost of m pens at $d each = __________________________________________________________b The perimeter of a square of side length l cm = _____________________________________________c If x is an odd number, the next odd number after x = __________________________________________d The distance travelled by a person at k km/h in h hours = ______________________________________e The number of minutes in T hours = _______________________________________________________

QUESTION 3 Write an algebraic expression for each of the following, using grouping symbols ifnecessary.

a Double k and divide the result by 15 = _____________________________________________________b Multiply 3a and 9b and to this result add 7 = _______________________________________________c Eight times the sum of 5x and 11y = _______________________________________________________d Add 14 to 3x and multiply the result by 9 = _________________________________________________e The product of a and 2b + 3c subtracted from 9x = ___________________________________________

QUESTION 4 Write an algebraic expression for the following.a 2x is divided by 3y and z is added to it = _______ b The number of metres in k kilometres = ______

________________________________________ _______________________________________c The number of km in M metres = ____________ d The number of grams in Y kilograms = _______

________________________________________ _______________________________________e The number of millimetres in x metres = ______ f The number of hours in s seconds = _________

________________________________________ _______________________________________

QUESTION 5 Explain the difference between each pair of algebraic expressions, then find the valueof each when m = 3 and n = 5:

a m2 and 2m b m3 and 3m c 2m2 and (2m)2______________________ ______________________ ____________________________________________ ______________________ ______________________

d 3m2 and 1

3m2 e m2n and mn2 f m2 + n2 and (m + n)2______________________ ______________________ ____________________________________________ ______________________ ______________________

CHAPTER 2Algebraic techniques

UNIT 1: Algebraic expressions

Chapter 2: Algebraic techniques 11

ANSWERS

page 149



Topic 2: Addition and subtraction in algebra

QUESTION 1 Add the following expressions.a 4x + 12x = ________________________ b 7x + 11x = ________________________c 9x + 8x = ________________________ d 20x + 14x + 3x = ________________________e 15x + 28x = ________________________ f 5a + 7a + 9a = ________________________g 5ab + 10ab + 2ab = _______________________ h 7mn + 2mn + mn = ______________________i 6p + 3p + 9p = ________________________ j 8x2 + 9x2 + 3x2 = ______________________k 15a2 + 6a2 + 2a2 = ________________________ l 5n + 8n + 10n + n = _____________________

QUESTION 2 Subtract the following expressions.a 18a – 3a = ________________________ b 9x – 8x = ________________________c 17y – 12y = ________________________ d 10m – 3m – m = ________________________e 15x – 4x – 5x = ________________________ f 7xy – 2xy – xy = ________________________g 14x2 – 5x2 – 3x2 – x2 = ____________________ h 16n – 4n – n – 2n = _____________________i 12p – 3p – p – 2p = ______________________ j 10a2 – 2a2 – a2 – 4a2 = ___________________k 8y – 3y – y – y = ________________________ l 9x – 7x – x – 2x – 3x = ___________________

QUESTION 3 Simplify the following expressions by adding or subtracting.a 10a + 5a – 4a = ________________________ b 7x + 8x – 3x – 6x = _____________________c 16a – 4a + 12a – 8a = ____________________ d 8mn – 3mn + 2mn = _____________________e 5p2 + 7p2 – p2 – 2p2 = ____________________ f 16ab + 8ab – 7ab – ab = _________________g 9t + 7t + 6t – 15t = ____________________ h 15a + 7a – a – 2a = _____________________i 4m2 – 3m2 + 8m2 – m2 = __________________ j 16t + 8t – 7t – t = ___________________k 9x – 3x + 2x – x = _______________________ l 8mn + 6nm – 5mn = ___________________

QUESTION 4 Simplify the following.a 8a + 3b – 5a + b = _____________________ b 16x + 4x – 5y + 7y = _____________________c 18a2 + 9a2 – 5b2 – b2 = __________________ d 14m + 5n – 3m – 2n = ____________________e 6a + 9b + 3b – 5a = _____________________ f 8m + 3n – 2n – 6m = ____________________g 14x + 5x – 6y – 3y = ____________________ h 9p + 7q – p – q = _____________________i 16ab2 + 3a2b – ab2 – 2a2b = _______________ j 9x + 3x – 2y – 6y = ___________________

QUESTION 5 Simplify the following expressions.a 25 – 12x + 8x – 7 = _____________________ b 8x2 + 7x2 – 9y2 = _____________________c 15a + 7b – 8a = _____________________ d 9m + 4n – 6m – n = ____________________e 20x + 4y – 6x – 2y = _____________________ f 16xy + 4yx – 7yz – yz = _________________g 15p + 8p – 9q = _____________________ h 5ab + 3ba + 9ab – ab = __________________i 14t + 10 – 6t – 12 = _____________________ j 4xy + 9yz – 3yx – 8zy = __________________

Algebraic techniquesUNIT 2: Addition and subtraction in algebra

12 EXCEL ESSENTIAL SKILLS: YEAR 10 MATHEMATICS REVISION AND EXAM WORKBOOK 1

ANSWERS

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Topic 3: Multiplication and division in algebra

QUESTION 1 Multiply the following.a 5 � 4x = ______________ b 6x � 4y = ______________ c 4xy � 5x = ______________d 9y � y = ______________ e –6m � 3n = ______________ f 2p � 3q = ______________g –6a2 � 5a = ______________ h 9x3 � (–2x) = ____________ i 5ab � (–4ba) = __________j –2 � 4x � –3y = __________ k (–xy) � (–yz) = ___________ l –20xy � (– 15yx) = ________

QUESTION 2 Divide the following expressions.a 14x � 7 = ______________ b 18y � 6y = ______________ c 10x2 � 10 = ____________d 36mn � 9m = _____________ e 12ab � ab = ______________ f 27 � 9x = ______________g –3xy � x = ______________ h –48a � 6a = ______________ i 2xyz � xy = ____________j 15xy � –3x = _____________ k –28ab � –7a = ____________ l –64abc � 16b = __________

QUESTION 3 Work out the following divisions.a 36ab � (–4a) = ___________________________ b 12xy � (–12xy) = ________________________c m2n2 � mn � m = _________________________ d 18xy � 2x = ____________________________e 14a2b2 � 7abc = __________________________ f 24x � 8x � x = _________________________g 40x2y2 � 10xy �2y = ______________________ h 26abc � ac � 26 = ______________________i 21 � 14ab = _____________________________ j 27a2 � (–9a) = _________________________

QUESTION 4 Simplify the following expressions.a 5 � 3k � 2ky = __________________________ b 4x � 2y � 3z = __________________________c 14x � 3 � 2x = __________________________ d 10x � 5 � 3x = __________________________e 18xy � xy � 18 = ________________________ f 9m � 7n � 3n = _________________________g 8x � 9y � 3x = __________________________ h 42a2b2c2 � 7abc � 2 = ___________________i a2b � ab � 3a = ________________________ j 16xy � 5x � 8y = ________________________

QUESTION 5 Simplify the following.a xy � 8yz � 4xz = _________________________ b 15am � 5m � 3a = _______________________c 9x � 3 � 2x = ___________________________ d 14xy � 2x � 4y = ________________________e 14a2 � 4a � 2a = ________________________ f 4 � 6xy � xy = _________________________

g 10x � 5y = _____________________________ h 14a � 5b = _____________________________25xy 7a2b

i (4a)2 � (5b)2 = __________________________40ab

Algebraic techniquesUNIT 3: Multiplication and division in algebra


ANSWERS

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Topic 4: Index notation and algebra

QUESTION 1 Simplify the following products.a x6 � x4 = ______________ b a5 � a7 = ______________ c m3 � m8 = ______________d y � y6 = ______________ e n4 � n11 = ______________ f a3 � a4 � a2 = __________g 4x5 � 7x3 = ______________ h 6a9 � 4a7 = ____________ i 3a5 � 4a6 = ____________j m7 � m5 � m4 = ___________ k a4b5 � a3b2 = ___________l 5xy � 4x2y3 � 3x3y3 = ___________________QUESTION 2 Simplify these divisions.a a9 � a6 = ______________ b x10 � x4 = ______________ c n15 � n11 = _____________d 24x7 � 8x4 = _____________ e y9 � y6 � y = ____________ f n7 � n3 � n = ___________g 36y12 � 12y5 = ____________ h 25x8 � 5x6 = _____________ i 30a4b5 � 15a2b2 = _______j x8y5 � x4y4 = _____________ k 32x9 � 8x7 = _____________ l 20x16 � 5x12 = __________

QUESTION 3 Simplify these expressions.a (a2)4 = ______________ b (x7)5 = ______________ c (y4)2 = ______________d (m3)2 = ______________ e (n4)7 = ______________ f (4p3)2 = ______________g a0 = ______________ h (5y)0 = ______________ i (9x0)2 = ______________j x0 + 6y0 – (3z)0 = __________ k 7(x2)0 � 2(y4)0 = __________ l –12p0 = ______________

QUESTION 4 Use the index laws to simplify the following.a 8a4b3 � 6a2b4 = __________ b 9t6 � 5t4y2 = __________ c a3 � a4 � a5 = __________d a3x � a4x = ______________ e a12b8 � a9b5 = __________ f m10n9 � m7n5 =

__________g x5y9 � x3y4 = ___________ h x9m � x4m = __________ i e6x � e4x = ___________

QUESTION 5 Simplify the following.a 5(a3)2 � 6a0 = ___________ b (4a2b2)3 � 16a2b2 = ________ c a24 � a12 � a4 =

_________d (6a2)2 � 8a4 = ____________ e (12a)2 � (6b)2 = __________ f 12ab � 4a � b = _________g (x7)2 � x5 = ____________ h a3b2 � a2b2 � ab = ________ i 64x2y2z � 16x2y2 = _______

QUESTION 6 Use the index laws to simplify the following.a 3x3y2 � 4x5y7 = _________________________ b 7a7b3 � 7a6b4 = _________________________c (7x2y3)0 = ______________________________ d (9xy)0 � 9(xy)0 = ________________________e 14a6 � 7a4 = ___________________________ f a7b4 � (a2b)3 � a6 = _____________________

g (5a2b3)2 = ______________ h (3x4)3 = _______________ i (a3 )2 � (5a)3 = ________c2 y2 2b3 3b

Algebraic techniquesUNIT 4: Index notation and algebra


ANSWERS

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Topic 5: Revising and combining index laws

QUESTION 1 Simplify.a (x3)2 = ______________ b (x2)3 = ______________ c (a4)5 = ______________y y3 b3

d (m8)2 = ______________ e (a5)2 = ______________ f (m8)2 = ______________m3 b7 n7

g (m6)4 = ______________ h (x2)4 = ______________ i (a9)3 = ______________2 y3 b2

QUESTION 2 Simplify.a (4a3)2 � a5 = ____________ b (2x2)3 � x9 = ____________ c (5n3)2 � n8 = ____________d (7t3)2 � t4 = ____________ e (a3)4 � (2a3)2 = ___________ f (x4y5)2 � (xy)5 = _________g (p2q)5 � (p3q4)2 = _________ h (2a4b3)2 � (6ab)2 = ________ i (5x2y3)2 � (2xy)4 = _______

QUESTION 3 Simplify.a 5a0 � (4a)0 = ______________ b 8x0 � (8x)0 = ______________ c (9y)0 � 9y0 = ____________

d (5x)0 = ______________ e (7p)0 � (5q)0 = ____________ f (6m)0 = ____________5x0 6m0

g 8x0 � (4x)0 � 5y0 = ________ h 8a0 = ______________ i 9x0 � (4x)0 � 7x0 = _______(5a)0

QUESTION 4 Simplify.a 9a � 6a2 � 4a3 = __________ b 32x5 � 8x4 � 5x = _______ c (5x)4 � 5x2 = ___________

d (7a2)3 = ______________ e 12m4 � 18m3 = ___________ f (6a2)3 � (8a)2 = _________(7a3)2 9m2 � 4m5 3a

g (8c2)3 = ______________ h 9m6 � 8m9 = _____________ i (4m5)3 �(2n)3 = ________4c2 � 6c3 24m8 3 64m8

QUESTION 5 Write in simplest form.a (a4)2 � a5 = ______________ b (y3)3 � y8 = ______________ c ( a )3 �

b8 = __________3 4 b2 a4

d ( x )5 �x4 = ____________ e (m4)3 �

m8 = ______________ f ( 3 )2 �1 = ___________y3 y6 n2 n4 t5 t8

g (m4)2� (n4)2 = ___________ h (6y2)3 � (4y)2 = ___________ i 96m12 � 8m4 � 6m2 =

(n3)3 m 8y

QUESTION 6 Simplify.a (m2n2p)4 � m2n = __________ b 48x8y6 = ______________ c m6n8 � (m2n)3 = _________8x4y4 (mn)4

d 8 � 8x0 + (8x)0 = __________ e (6a4)2 = ___________ f 9y4 � 6y8 = _____________(6a2)4 3y6 � y2

g 20x6 � (2x3)2 = ___________ h (9k2)3 � (4k2)2 = __________ i (5y5)3 = __________10x8 3k 5y2 � (5y)2

Algebraic techniquesUNIT 5: Revising and combining index laws


ANSWERS

page 149



Topic 6: Negative indices

QUESTION 1 Evaluate the following.a 2–1 = ______________ b 3–2 = ______________ c 4–3 = ______________d 5–2 = ______________ e 6–3 = ______________ f 7–1 = ______________g 8–2 = ______________ h 9–3 = ______________ i 6–2 = ______________j 3–4 = ______________ k 2–6 = ______________ l 5–4 = ______________

QUESTION 2 Write the following with positive indices.a 3–4 = ______________ b m–8 = ______________ c x–5 = ______________d (–5)–3 = ______________ e 1 = ______________ f (79)–2 = ______________2–5

QUESTION 3 Write the following with negative indices.

a 1 = ______________ b 1 = ______________ c 7 = ______________32 53 x8d 1 = ______________ e 8a = ______________ f 1 = ______________6m3 15b3 7x5

QUESTION 4 Simplify, writing your answers with positive indices.a x–3 � x8 = ______________ b a–12 = ______________ c 9m–5 = ______________d 64a–9 � 32a–6 = ___________ e (a–7)4 = ______________ f 72m8 � 9m6 = ___________

QUESTION 5 Simplify, expressing your answers with positive indices.a 5a � 3a–4 = ____________ b 12t–6 � 6t–3 = ____________ c x–8 � x� x–3 = __________d 28a–5 � 7a3 = ____________ e (x–2)–4 = _____________ f (3y2)–3 = ___________g (x2y2z2)–5 = ____________ h a–4b–7 = ____________ i (a8)–2 = __________b5

QUESTION 6 Given that a = 2, b = 4, c = 12 and d = 1

4, evaluate.a a–2 = ____________ b a–2b–2 = __________ c a–1 � b–2 = __________d c–3b2 = ____________ e a–1b–1c–2d–2 = __________ f (ab)–2 � (cd)–2 = _________

QUESTION 7 Calculate the value of x in.

a 3–2 = 3x ____________ b 8–4 = 1 __________ c 5–3 �1 __________8x 5x

d 4–9 � 45 = 4x ____________ e 1� 33 = 33x __________ f 1 = 9x _________3–6 27

Algebraic techniquesUNIT 6: Negative indices


ANSWERS

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Topic 7: Fractional indices

QUESTION 1 Express in surd form.

a 25 = ______________ b 64 = ______________ c a = ______________

d 8 = ______________ e (5x) = ______________ f (125m) = ______________

QUESTION 2 Express in index form.

a ��11 = ______________ b ��x = ______________ c 3�216a ______________d 7��65 = ______________ e n��20 = ______________ f 1 = ______________��21

QUESTION 3 Evaluate the following.

a 64 = ______________ b (512) = ______________ c (216) = _____________

d (49a6) = ____________ e (a8b8) = ______________ f (27y12) = _____________

QUESTION 4 Simplify.

a a � a = ______________ b 5m � 6m = ______________ c x � x = _____________

d (y ) = ____________ e (a14b10) = ______________ f (243a20) = _____________

QUESTION 5 Simplify the following.

a x � x4 � x = __________ b y � y = _______________ c (a ) = _____________d 8a � 5a + 2a = ____________ e m–4 � m = ______________ f ( 3 ) = _____________

p2

QUESTION 6 Write in simplest form.

a a � a = ___________ b (a–2b4) = ___________ c (a3) � (a )3 = _________

d a � a � a– = ___________ e 5��a4 �4��a5 = ___________ f (x–4y3)–3 = _____________

QUESTION 7 Given that a = 8, b = 16 and c = 64, evaluate:a a � b = ___________ b (bc) = ___________ c a � c = ____________

d a � b = ___________ e a � b = ___________ f (ab) � a0 = ___________

Algebraic techniquesUNIT 7: Fractional indices


ANSWERS

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13

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– 13

– 27– 13 – 14

23

13

13

16

13

12

13

23

49

14

12

14

12

12

23

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23

23

23

14

78

43

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87

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Topic 8: Grouping symbols in algebra

QUESTION 1 Expand the following expressions.a 3(x + 2) = ________________________ b 2(a + 5) = ________________________c 4(2y – 1) = ________________________ d 3(6a + 7) = ________________________e 5(8 – a) = ________________________ f 6(2k – 3) = ________________________g 5n(n – 1) = _______________________ h 3(4 – 3a) = ________________________i 7(2n + 7) = ________________________ j y(2y + 7) = ________________________k m(m + 10) = ________________________ l 2a(3a – 7) = ________________________

QUESTION 2 Remove the grouping symbols.a –2(2a + 3) = ________________________ b –3(5n – 4) = ________________________c –(y + 8) = ________________________ d –5(7 + 2t) = ________________________e –3(5x + 18) = ________________________ f –4(3x – 2) = ________________________g –(6x + 11) = _______________________ h –2(4x – 9) = ________________________i –5(4x – 5) = ________________________ j –3(a – 14) = ________________________k –8(x – 10) = ________________________ l –(2 – 5x) = ________________________

QUESTION 3 Expand the following expressions.a 1

3(9x – 15) = ________________________ b 12(8x – 4) = ________________________

c –14(24y – 8) = ________________________ d a3(2a + 3) = ________________________e a2(3a + 4b) = ________________________ f –2y(3y + 7) = ________________________g –y2(3y – 6) = _______________________ h 4t2(5t – 8) = ________________________i –m(3m2 + 5m) = ________________________ j –6p(3p2 + 5) = ________________________k –4x(8x – 1) = ________________________ l 3n(8n2 +7n) = ________________________

QUESTION 4 Expand.a –2(5x + y – z) = ________________________ b –3(2a + 3b – 4c) = ______________________c 4(a2 – 3a + 7) = ________________________ d –(5t2 – 3t + 4) = ________________________e 3(2xy + 3xy2 – 8x) = _____________________ f 2ab(4a2b – 6ab + 3ab2) = _________________g –5a(3a – 2b + 4c) = _____________________ h 3p(8p – 2q + 3r) = _____________________i 4a(a2 + 2ab – 3ac) = ____________________ j –a(2a + 3b – 9c) = _____________________k –t(2t3 + 3t2 – 5t) = ________________________ l 8(9x – 7y + 2z) = _______________________

QUESTION 5 Expand.a 3t(t4 – 5t3 + 2t2 – 8t – 7) = ________________ b m(5m4 – 3m3 + 2m2 – m – 1) = ____________c x2(4y2 – 3xy + 4x – 7y) = __________________ d ab(a4 – a3 + 4ab – 2a2 – 3ab2) = ___________e –4a(5a3 + 4a2 + 3a – 2) = __________________ f –2y(8y2 + 7y – xy + 6) = _________________g –ab(a3 + b2 – 2ab + c) = __________________ h –4x(x3 + y2 – 2xy – x) = __________________

Algebraic techniquesUNIT 8: Grouping symbols in algebra


ANSWERS

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Topic 9: Expanding and simplifying algebraic expressions

QUESTION 1 Expand and simplify.a 5(x + 3) + 2x – 5 = ___________________ b 3(a + 2) + 2a – 7 = ___________________c 7(2m – 1) + 10m – 3 = ___________________ d 6a + 7 – 2(2a + 4) = ___________________e 8y – 3 – 2(y + 5) = ___________________ f 9x + 2(3x – 1) + 6 = ___________________g 5t + 6 + 3(t2 + 5) = ___________________ h 5x – (2x – 1) + 3x = ___________________i 18 – 2(x – 2) + 4x = ___________________ j 7(2m – 5) – 4m + 1 = ___________________k 8a + 7 – 2(4a – 1) = ___________________ l 7x + 11 – 2(x – 3) = ___________________

QUESTION 2 Remove the grouping symbols and simplify.a 5(2a + 4) + 3 = ___________________ b 7(2t – 7) + 5t = ___________________c 6m + 3(2m – 5) = ___________________ d 9p + 2(8 – 3p) = ___________________e 10y + 3(8y – 1) = ___________________ f 6(3x – 10) + 5x = ___________________g 7(3 – n) – 9n = ___________________ h 9y(y + 3) – 4 = ___________________i 6a – 4(2a – 3) = ___________________ j 25 – 2(4x – 5) = ___________________k 9x – (3x – 2y + z) = ___________________ l 5t + 3(9 – 2t) – 8 = ___________________

QUESTION 3 Simplify.a 2(x + 3) + 4(x – 1) = ____________________________________________________________________b 5a(a2 – 2a – 3) – a(a + 9) = ______________________________________________________________c 3xy(x2 – y – 7) – x2(x + 3) = _____________________________________________________________d 5(m + 3n) – 3(2m – 6n) – 2(m + 8) = _______________________________________________________e 2t(t2 – 3t + 3) – 5t(3t2 – 2t – 1) = __________________________________________________________f 7a4 – 5a3 + 2a2 – 3a – 2(10 – 5a + 3a2) = ___________________________________________________

QUESTION 4 Write in simplest form.a Add 2a + 3b to 7a – 5b = ________________________________________________________________b Add 5x – 3y + z to 8x + 5y – 3z = __________________________________________________________c Find the sum of 2m + 3, 9 – 5m and m – 10 = ________________________________________________d Subtract 5a – 7 from 18a – 10 = ___________________________________________________________e Subtract y2 – 4y + 6 from 4y2 – 10y + 9 = ___________________________________________________f From 8t2 – 5t – 9 take 5t2 + 4t – 3 =________________________________________________________

QUESTION 5 Simplify each expression to check that the given solution is correct.a 3(x + 5) + 2(2x + 3) = 7x + 21 ____________________________________________________________b 9(a – 7) – (2a + 1) = 7a – 64 _____________________________________________________________c 8(2y – 4) – 3(6y – 10) = –2y – 2 ___________________________________________________________d 5(3 – t) – 2(2t + 1) = 13 – 9t ______________________________________________________________e (8x – 7) – (4x – 3) – (x – 2) = 3x – 2 ________________________________________________________f 5(2a + 3b + c) – (a + b – c) = 9a + 14b + 6c _________________________________________________

Algebraic techniquesUNIT 9: Expanding and simplifying algebraic expressions


ANSWERS

page 150



Topic 10: Substitution

QUESTION 1 Calculate the value of each expression given that a = –2, b = 3 and c = 4.a a + b = ________________________ b a + b + c = ________________________c b + c = ________________________ d c + a = ________________________e a + b – c = ________________________ f a – b + c = ________________________g 3a + 2b = _______________________ h 4b – 5c = ________________________i a + 2b + 3c = ________________________ j a2 + b2 = ________________________k a2b + b2a = ________________________ l a + b = ________________________ab

QUESTION 2 If x = 3, calculate the value of the following expressions.a 4x2 = ________________________ b (4x)2 = ________________________c 30 – 5x = ________________________ d (6x – 7)2 = ________________________e (x – 1)(x – 8) = ________________________ f �x2 – 5 = ________________________g (x – 2)3 = _______________________ h 4x2 � 5x = ________________________i (x + 2)(x – 2) = ________________________ j 20 – x2 = ________________________k 5x2 – 8x = ________________________ l x2 + 4x – 6 = ________________________

QUESTION 3 If x = 12 and y = 1

3, find the value of.a x + y = ________________________ b x – y = ________________________c x + y = ________________________ d x – y = ________________________x – y x + y

e x + y + x – y = ________________________ f x2 + y2 ________________________x – y x + y

g x2 – y2 = _______________________ h xy = ________________________x + y

i x – y = ________________________ j (x + y)2 = ________________________xy

k (x – y)2 = ________________________ l x + y = ________________________y x

QUESTION 4 Given that x = 8·5, y = 5·2 and z = 6·4, find correct to one decimal place the value of:a xy2 = ________________________ b x2y = ______________________c xy + yz = ________________________ d (x + y)2 = ________________________e (x + y)(x – y) = _____________________ f �x + y + z = _________________g xyz � 3 = _____________________ h x + y = _____________________y + z

i x + y = ____________________ j (2x + 3y)2 = _____________________y z

Algebraic techniquesUNIT 10: Substitution


ANSWERS

page 150



Topic 11: Addition and subtraction of algebraic fractions

QUESTION 1 Find the sum of these algebraic fractions.

a m + 3m = ______________ b x + 2x = ______________ c 2t + 3t = ______________5 5 3 3 7 7d 5y + 3y = ______________ e 6a + 3b = ______________ f 16 + 4 = ______________8 8 11 11 5a 5ag 19k + 14k = ______________ h 2a + 8a = ____________ i 15x + 2x = _____________8 8 5x 5x 17 17j 10p + 4p = ___________ k 8a + 9a = ___________ l 6m + 3m = _____________7 7 5b 5b 19 19

QUESTION 2 Subtract the following algebraic expressions.

a 12y – 9y = ______________ b 5x – 3x = ______________ c 8a – 5a = ______________7 7 11 11 9 9d 12a – 5a = ______________ e 5a – 4a = ______________ f 14 – 9 = ______________17 17 7x 7x 5t 5tg 9m – 7m = ______________ h 16 – 10 = ____________ i 5m – 3m = _____________23 23 3x2 3x2 12 12j 9a – 5a = ___________ k 12x – 7x = ___________ l 5a – 2b = _____________7 7 10 10 11 11


a 2a�

a = ______________ b 2x�x = ______________ c 5y

�3y = _____________5 10 3 6 8 4

d 2k�3k = ______________ e y

�y = ______________ f x

�3x = ____________5 15 8 32 4 8

g 8y�3y = ______________ h 5p

�p = ____________ i 3t

�2t = _____________3 5 8 4 2 21

j 9�3 = ___________ k 5a

�3a = ___________ l 8m

�3m = __________x 4x 7b 14b 5n 20n


a 3x�

x = ____________ b 4m�2m = ___________ c 3a

�2b = __________5 10 7 21 8 4

d 5t�

t = ______________ e 8y�3y = _____________ f 7m

�4m = ___________8 4 3 4 12 3

g 2a� 3�a � 1 = _________ h 5x – 3

�2x � 1 = ________ i 8m � 3

�3m � 4 = _______7 7 8 8 5 5

j 2x � 3�x � 1 = _________ k 2y – 3

�y – 1 = _________ l 4t – 3

�2t � 1 = ________5 4 2 2 4 8

Algebraic techniquesUNIT 11: Addition and subtraction of algebraic fractions


ANSWERS

page 150



Topic 12: Multiplication and division of algebraic fractions

QUESTION 1 Find the products of these algebraic fractions.

a a�b = ______________ b m

�n = ______________ c x

�y = ______________5 7 2 6 3 8

d 5x�4y = ______________ e a

�a = ______________ f 5

�3 = ______________3 9 3 11 y t

g 4�5 = ______________ h 6

�3 = ____________ i 8

�4 = _____________a b 5a 2b 3t 2t

j 2x�3x = ___________ k 4

�2a = ___________ l 5b

�2b = _____________3y 2y m 3n 3c 9c

QUESTION 2 Divide the following algebraic fractions.

a a�

a = ______________ b 3n�5n = ______________ c p

�3p = ______________5 15 8 16 2n 8n

d 8�5 = ______________ e 7

�14 = ______________ f a

�5a = ______________x x 2y 10y b b

g 9n�27n = ______________ h xy

�z = ____________ i 15t

�5t = _____________5m 15m z xy m 7m

j 20p�10p = ___________ k k

�km = ___________ l xyz

�yz = _____________11 22 18 36 15 5


a 3t�10 = ______________ b 5x

�5 = ______________ c 15x

�33y = ____________20 24t 7 7 11y 60x

d 30n�4m2 = ______________ e 8ab

�ac = ______________ f 24a

�34b = ____________m3 15n c 4b 17b 16a

g x�y

�z = ______________ h x2y

�m = ____________ i 4ab

�10c = _____________y z x m xy 5c 8a

j 6ab�

x3�10 = ___________k ab

�ac

�bc = __________ l 15x

�32x2y = __________x2 5ab x bc ab ac 8y 25x3y2


a a3b2�ab2 = _____________ b m2

�n2

�p2 = ___________ c xyz

�pq2 = __________ab c3 n2 p2 m2 p2q2 x2y2z2

d pq�

q = ______________ e ab�a3b2 = ______________ f x3y3

�x3y2 = _________

20 40 c3 c4 z2 z3

g 4a2b3�16a3b2 = __________ h 18mn

�48m = ___________ i 20ab

�4a = _____________7 21 11p 33p 27 36

j –3ab�–12a2b2 = __________ k –8a

�–48ab = __________ l 35mn

�7m2 = _________9c 45c2 9b 18c 6p 12p

Algebraic techniquesUNIT 12: Multiplication and division of algebraic fractions


ANSWERS

page 150



Topic 13: Factorisation using common factors

QUESTION 1 Factorise the following by taking the common factor out.

a 4x + 16 = ______________ b 9a � 27 = ______________ c 5x � 25x2 = _____________d 7a � 21a2b = _____________ e 5ab � 25a2b2 = ___________ f 7m � 21m2n = ___________g a3b2 � a2b3 = ____________ h 14x3y3 � 28x2y2 = _________ i 15ab � 25bc = __________j 12ab � 15a2 = ___________ k x2y2 � 7xy = ___________ l abc � 6bcd = ___________

QUESTION 2 Factorise the following by taking the negative common factor out.

a –4a � 28 = ______________ b –3a � 15 = ______________ c –8x � 32 = ____________d –10xy � 15y = _____________ e –8y � 40 = ______________ f –m3 � m2 = ____________g –x3 � 10x2y2 = ____________ h –6x2 � 12x = _____________ i –10y2 � 12y = ___________j –5x � 9x2 = ______________ k –3m � 18m3 = ___________ l –9m � 36m4 = ___________

QUESTION 3 Factorise the following.

a a(a + 2) � b(a + 2) = ____________________ b 3(x + y) � a(x + y) = ____________________c 9(x – y) + 2a(x – y) = ____________________ d 5(2a + 3b) – c(2a + 3b) = _________________e x2(5 – y) – 3(5 – y)= _____________________ f x(2x – 9) + 5(2x – 9) = ___________________g m(a – b) – n(a – b) = ______________________ h 12(x2 + 7) – y(x2 + 7) = __________________i 5(x + 8) + y(x + 8) = ______________________ j 4a(3b – 5c) + 2(3b – 5c) = ________________k m(2n – p) – q(2n – p) = ____________________ l 3x2(2a – 5b) + y2(2a – 5b) = ______________

QUESTION 4 Factorise each of the following.

a mx � my � mz = _________________________ b ac � bc � cd = _________________________c 5m � mn � 6mp = ________________________ d 10a � 25b � 35c = ______________________e 20xy � 8x2 � 36 = ________________________ f n2 � 8mn � 10n = _______________________g 5a2 � 15abc � 10a = ______________________ h xy2 � 2xy � x2y = _______________________i 3a � 9ab � 15a2 = ________________________ j 5m � 10mn � 20m2n = ___________________k x3y2 � 2x2y2 � 3x2y3 = ____________________ l 12x2y2z2 � x3y2 � x2y3 = _________________

Algebraic techniquesUNIT 13: Factorisation using common factors

Chapter 2: Algebraic Techniques 23

ANSWERS

page 150

Total marks achieved for PART A 15

Algebraic techniquesTopic Test PART A

Marks

14Algebraic techniques

Mathletics Instant Workbooks – Series K Copyright © 3P Learning

TOPIC TEST PART AAlgebraic techniquesTime allowed: 15 minutes Total marks = 15

1 a3 � a3 equals:�A 2a3 �B 2a6 �C a6 �D a9

2 15x10 � 5x5 equals:�A 3x5 �B 3x2 �C 10x5 �D 10x2

3 (4m3)2 equals:�A 8m5 �B 8m6 �C 16m9 �D 16m6

4 (5y3)0 equals:�A 5y3 �B 5 �C 0 �D 1

5 4x–2 equals:�A 1

�B 4�C –1

�D –44x2 x2 4x2 x2

6 m � m equals:�A m �B m �C m �D 2m

7 (25a2b4) equals:�A 5ab2 �B 25ab2 �C 5a3b3 �D 25a3b3

8 If x = –3 and y = 2, the value of xy2 is equal to:�A –12 �B 12 �C –18 �D 18

9 5a – (2 – a) equals:�A 4a – 2 �B 6a – 2 �C 5a – 2 �D 4a + 2

10 3(x + 4) + 2 equals:�A 3x + 6 �B 3x + 18 �C 3x + 9 �D 3x + 14

11 If x = –2 the –x3 equals:�A 8 �B –8 �C 6 �D –6

12 If x is an integer, which of the following will produce an odd number?�A x2 �B 3x2 �C 3x2 + 1 �D 3x2 + 2x

13 5x – (–x) equals:�A 5 �B 5x �C 5x2 �D 6x

14 The correct factorisation of 3xy – x is:�A 3x(y – 1) �B 3x(y – x) �C x(3y – 1) �D x(3y – x)

15 15a2 equals:5ab�A 3ab �B 3a

�C 3b �D 3b b

Total marks for PART A 15

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

ANSWERS

page 150

12

13

1316

19

23

23


TOPIC TEST PART AAlgebraic techniquesTime allowed: 15 minutes Total marks = 15

1 a3 � a3 equals:�A 2a3 �B 2a6 �C a6 �D a9

2 15x10 � 5x5 equals:�A 3x5 �B 3x2 �C 10x5 �D 10x2

3 (4m3)2 equals:�A 8m5 �B 8m6 �C 16m9 �D 16m6

4 (5y3)0 equals:�A 5y3 �B 5 �C 0 �D 1

5 4x–2 equals:�A 1

�B 4�C –1

�D –44x2 x2 4x2 x2

6 m � m equals:�A m �B m �C m �D 2m

7 (25a2b4) equals:�A 5ab2 �B 25ab2 �C 5a3b3 �D 25a3b3

8 If x = –3 and y = 2, the value of xy2 is equal to:�A –12 �B 12 �C –18 �D 18

9 5a – (2 – a) equals:�A 4a – 2 �B 6a – 2 �C 5a – 2 �D 4a + 2

10 3(x + 4) + 2 equals:�A 3x + 6 �B 3x + 18 �C 3x + 9 �D 3x + 14

11 If x = –2 the –x3 equals:�A 8 �B –8 �C 6 �D –6

12 If x is an integer, which of the following will produce an odd number?�A x2 �B 3x2 �C 3x2 + 1 �D 3x2 + 2x

13 5x – (–x) equals:�A 5 �B 5x �C 5x2 �D 6x

14 The correct factorisation of 3xy – x is:�A 3x(y – 1) �B 3x(y – x) �C x(3y – 1) �D x(3y – x)

15 15a2 equals:5ab�A 3ab �B 3a

�C 3b �D 3b b

Total marks for PART A 15

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

ANSWERS

page 150

12

13

1316

19

23

23


Time allowed: 15 minutes Total marks = 15

Algebraic techniquesTopic Test PART B

15Algebraic techniques

Mathletics Instant Workbooks – Series K Copyright © 3P Learning

Total marks achieved for PART B 15

Marks

Time allowed: 15 minutes Total marks = 15

Answers

TOPIC TEST PART BAlgebraic techniquesTime allowed: 15 minutes Total marks = 15

Question 1 Simplify the following.a (5x3)2 =

___________________________b 52m8 � 13m6 =

___________________________c (49x16) =

___________________________d 8y0 + (8y)0 + (8y8)0 =

___________________________e 8 � 8 � 36 � 36 =

___________________________

Question 2

a Simplify 2a + 2b – 2c – (2a + 2b + 2c)= ___________________________b Expand 34(16xy + 32x2 – 12y2) = ___________________________c Remove the grouping symbols and simplify

4(5x – 3) – 2(3x + 8). ___________________________d Simplify 5(3a – 7) – 4(2 – 8a) = ___________________________e Find the sum of 3a + 2b – 9 and 6a – 5b – 3. ___________________________

Question 3

a Simplify (a–3b–3)–3 � ( 1 )3 = ___________________________a3

b Simplify 5m �3m = ___________________________4 8

c Simplify a3b4 �a2b2 = ___________________________c2 c3

d Given that a = 12, b = 1

3, find the value of (ab)2 + (a + b)2 ___________________________e Factorise 8a2 + 24ab – 16a = ___________________________

Total marks for PART B 15

1

1

1

1

1

11

111

1

1

1

1

1

ANSWERS

page 150

12


13

13

12

12

Answers Marks



___________________________b 52m8 � 13m6 =

___________________________c (49x16) =

___________________________d 8y0 + (8y)0 + (8y8)0 =

___________________________e 8 � 8 � 36 � 36 =

___________________________

Question 2



Question 3

a Simplify (a–3b–3)–3 � ( 1 )3 = ___________________________a3

b Simplify 5m �3m = ___________________________4 8





1

1

1

1

1

11

111

1

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1

1

1

ANSWERS

page 150

12


13

13

12

12

Answers Marks



___________________________b 52m8 � 13m6 =

___________________________c (49x16) =

___________________________d 8y0 + (8y)0 + (8y8)0 =

___________________________e 8 � 8 � 36 � 36 =

___________________________

Question 2



Question 3

a Simplify (a–3b–3)–3 � ( 1 )3 = ___________________________a3

b Simplify 5m �3m = ___________________________4 8





1

1

1

1

1

11

111

1

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1

1

1

ANSWERS

page 150

12


13

13

12

12

Answers Marks

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