indices...exercise task 1 1. 3if 2×2 Ô=20, what is the value of a? 2. if 4−3×4 Ô=1, what is...

Indices

Notes & Exercises

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Contents

Credit

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Facil

Multiplying (1) - Numerical

Multiplying (1) - Algebraic Page 4

Dividing (1) - Numerical Page 6

Dividing (1) - Algebraic Page 8

Power Zero Page 10

Everything so far… Page 12

Negative Exponents - Integers Page 13

Negative Exponents - Fractions Page 15

Negative Exponents - algebraic term on numerator Page 16

Negative Exponents - algebraic term on denominator Page 17

Multiplying (2) Page 18

Dividing (2) Page 19

Multiplying & Dividing (2) Page 20

Powers of Powers - numerical Page 21

Powers of Powers - algebraic Page 23


Expanding Single Brackets Involving Indices Page 30

Expanding Brackets Involving Indices Page 32

Unitary Fractional Indices Page 33

Non-Unitary Fractional Indices Page 38


Solving Exponential Equations Page 45

Changing Bases Page 46

Final Task Page 50


Multiplying (1) - Numerical

Explore

34 × 3

= (3 × 3 × 3 × 3) × 3

= 3___

34 × 32

= (___ × ___ × ___ × ___) × (___ × ___)

= 3___

34 × 33

= (___ × ___ × ___ × ___) × (___ × ___ × ___)

= 3___

34 × 3𝑛

= 3___

3𝑚 × 3𝑛

= 3___

Rewrite using a single exponent:

Example 1

95 × 92

Your turn…

86 × 83

Example 2

95 × 9−2

Your turn…

86 × 8−3


Task 1

Complete the following on the grids:

× 2 22 23 24 25 26

2

22

23

24

25

26

Rewrite the following with a single exponent in your jotter:

1. 34 × 32 2. 4 × 43 3. 103 × 102 4. 53 × 54

5. 35 × 35 6. 74 × 7−2 7. 27 × 2−3 8. 1010 × 10−9

9. 59 × 5−1 10. 3−4 × 39 11. (2

3)2

× (2

3)4

12. (−1

2)2

× (−1

2)3

13. (3

4)5

× (3

4)−2

14. (1

7)−7

× (1

7)20

15. 21

3 × 22

3 16. 154

3 × 152

3

17. 107

4 × 105

4 18. 30.4 × 30.2 19. 51.2 × 50.8 20. 51.2 × 5−0.8

Task 2

True or False? How do you know? Is there another way you can tell?

1. 23 × 32 = 65 2. 33 × 33 = 36 3. 23 × 32 = 66 4. 22 × 32 = 62

5. 23 × 22 = 45 6. 26 × 23 = 49 7. 63 × 64 = 612

Task 3

Investigate 23 × 53. Do you notice anything?

Can you generalise?

Rewrite the following and evaluate:

1. 23 × 43 2. 52 × 22 3. 142 × (1

2)2

4. (−5)3 × 23 5. 22 × 32 × 53 × 23

× 2−3 2−2 2−1

210

29

28


Multiplying (1) - Algebraic

Explore

𝑥3 × 𝑥2

= (𝑥 × 𝑥 × 𝑥) × (𝑥 × 𝑥)

= 𝑥___

𝑥3 × 𝑥3

= (___ × ___ × ___) × (___ × ___ × ___)

= 𝑥___

𝑥3 × 𝑥4

= 𝑥___

𝑥3 × 𝑥𝑛

= 𝑥___

𝑥𝑚 × 𝑥𝑛

= 𝑥___

× 𝑥 𝑥2 𝑥3 𝑥4 𝑥5 𝑥6

𝑥4

𝑥5

𝑥6

𝑥7

𝑥8

𝑥9

Task 1

Complete the following on the grids:

Simplify:

Example 1

𝑥7 × 𝑥8

Your turn…

𝑥9 × 𝑥2


Task 2

Simplify the following in your jotter:

1. 𝑎2 × 𝑎4 2. 𝑥 × 𝑥3 3. 𝑥2 × 𝑥6 4. 𝑥6 × 𝑥−2

5. 𝑎12 × 𝑎10 6. 𝑓23 × 𝑓−10 7. 𝑥7 × 𝑥8 8. 𝑦−6 × 𝑦7

9. 𝑏7 × 𝑏5 × 𝑏9 10. 𝑥2 × 𝑥 × 𝑥7 11. 𝑥4 × 𝑥5 × 𝑥6 12. 𝑥2 × 𝑥4 × 𝑥6 × 𝑥8

13. 𝑎2 × 𝑎4 × 𝑎−1 14. 𝑎3 × 𝑎−2 × 𝑎 15. 𝑥−2 × 𝑥−4 × 𝑥6 × 𝑥8

Task 3

Find as many pairs of values for m and n such that the statement below holds true:

𝑐𝑚 × 𝑐𝑛 = 𝑐8

× 𝑥−3 𝑥−2 𝑥−1

𝑥11

𝑥12

𝑥13

Simplify:

Example 2

3𝑥4 × 2𝑥5

Your turn…

4𝑥3 × 5𝑥7

Task 4

Simplify the following in your jotter:

1. 2𝑎2 × 𝑎3 2. 2𝑎4 × 3𝑎 3. 2𝑥6 × 3𝑥4

4. 5𝑥4 × 6𝑥2 5. 2

3𝑥3 × 12𝑥4 6. 2𝑎7 × 6𝑎2 ×

1

4𝑎

7. 4𝑎3 × 3𝑎2 × 5𝑎 8. −3𝑏5 × 8𝑏4 9. −3𝑐4 × −4𝑐9

10. 2𝑎4 × 3𝑎−1 11. 4𝑎5 × 8𝑎−3 12. 4𝑟−6 × 5𝑟7

13. 5𝑡7 × 2𝑡−4 × 3𝑡 14. 3𝑠2 × 4𝑠4 × −2𝑠6 15. 2

3𝑥4 ×−12𝑥3 ×

1

4𝑥−4

Task 5

If 2 3 4 2 2y yx x x x x x then:

A 0y B 1y C 1y D 1

2y


Dividing (1) - Numerical

Explore

24 ÷ 2 or 24

2

=2×2×2×2

2

= 2___

24 ÷ 22 or 24

22

=2×2×2×2

2×2

= 2___

24 ÷ 23 or 24

23

= 2___

24 ÷ 2𝑛 or 24

2𝑛

= 2___

2𝑚 ÷ 2𝑛 or 2𝑚

2𝑛

= 2___

Rewrite using a single exponent:

Example 1

95 ÷ 92

Your turn…

812 ÷ 83

Example 2

95 ÷ 9−2

Your turn…

812 ÷ 8−3


Exercise

Task 1

Rewrite the following using a single exponent in your jotter:

1. 54

52 2. 25 ÷ 23 3. 74 ÷ 73 4.

512

5

5. 0.37

0.33 6. 162 ÷ 16−1 7.

134

13−2 8. 0.28 ÷ 0.2−6

9. 9−4 ÷ 9−6 10. 6−3

6−12 11. (

1

2)18 ÷ (

1

2)13 12. (−

1

3)14 ÷ (−

1

3)10

13. (3

5)9 ÷ (

3

5)−3 14. (−

7

8)−2 ÷ (−

7

8)−7 15. 8

4

3 ÷ 81

3 16. 1994

1954

17. 2594

25−34

18. 43.1 ÷ 42.9 19. 151.4

15−0.6 20. 12−0.2 ÷ 12−0.9

Task 2

Simplify each quotient and then evaluate the result:

1. 106

102 2.

417

414 3.

9210

9207 4.

2𝑦+1

2𝑦 5.

8𝑟+4

8𝑟+1

Task 3

Spot the mistake(s)

715 ÷ 75 = 73

Task 4

True or False?

How do you know?

Is there another way you can tell?

1. 10−6 ÷ 10−8 = 10−14 2. 53 ÷ 23 = 33

Task 5

Given that 𝑝 = 5𝑚 and 𝑞 = 5𝑛 , write the following as a single power of 5:

𝑝

𝑞

Task 6

A formula is given as 𝐻 =2𝑎

4.

a) Calculate H when a = 6, can you express H as a power of 2?

b) Calculate a when H = 8.

c) Calculate the minimum value of H given 𝑎 ≥ 0.


Dividing (1) – Algebraic

Explore

𝑥5 ÷ 𝑥 or 𝑥5

𝑥

=𝑥×𝑥×𝑥×𝑥×𝑥

𝑥

= 𝑥___

𝑥5 ÷ 𝑥2 or 𝑥5

𝑥2

=𝑥×𝑥×𝑥×𝑥×𝑥

𝑥×𝑥

= 𝑥___

𝑥5 ÷ 𝑥3 or 𝑥5

𝑥3

= 𝑥___

𝑥5 ÷ 𝑥𝑛 or 𝑥5

𝑥𝑛

= 𝑥___

𝑥𝑚 ÷ 𝑥𝑛 or 𝑥𝑚

𝑥𝑛

= 𝑥___

Simplify:

Example 1

𝑦12 ÷ 𝑦4

Your turn…

𝑝14 ÷ 𝑝9

Exercise

Simplify

1. 𝑥6

𝑥 2.

𝑥13

𝑥2 3. 𝑐12 ÷ 𝑐4

4. 𝑥8 ÷ 𝑥3 5. 𝑥10

𝑥3 6. 𝑎12 ÷ 𝑎2

7. 𝑎12 ÷ 𝑎−2 8. 𝑡20

𝑡3 9.

𝑡20

𝑡−3

10. 𝑡2

𝑡−5 11.

𝑡−2

𝑡−5 12. 𝑏10 ÷ 𝑏−6

13. 𝑝−3 ÷ 𝑝−13 14. 𝑠14 ÷ 𝑠−6 15. 𝑎−2 ÷ 𝑎−5


Simplify:

Example 2 Your turn…

12𝑦11 ÷ 6𝑦7 56𝑦4

8𝑦2


5𝑦11 ÷ 12𝑦7 8𝑦4

56𝑦2

Exercise

Simplify

1. 6𝑥5 ÷ 3𝑥2 2. 3𝑥5 ÷ 6𝑥2 3. 6𝑥5 ÷ 3𝑥−2

4. 3𝑥5 ÷ 6𝑥−2 5. 20𝑥6

4𝑥5 6.

4𝑥6

20𝑥5

7. 36𝑥7

3𝑥4 8.

3𝑥7

36𝑥4 9.

36𝑥7

3𝑥−4

10. 3𝑥7

36𝑥−4 11.

1.3𝑥7

1.3𝑥4 12.

3

4𝑥5 ÷

3

4𝑥−2

13. 5.5𝑥−1 ÷ 1.1𝑥−5 14. 1.1𝑥−1

5.5𝑥−5 15.

2

3𝑏13 ÷

1

3𝑏3


The Power Zero

Explore 1 Explore 2

24 = 16 Any non-zero number divided by itself equals 1.

÷ 2 i.e. 2 ÷ 2 = 1

23 = 8 Using the exponent rule for division

÷ 2 21 ÷ 21 = 21−1 = 20 = 1

22 = 4

÷ 2 Can you generalise?

21 = 2

÷ 2

20 = 1

Try these…

Simplify:

1. 30 2. (−5)0 3. 1,000,0000 4. 𝑥0

Simplify:

Example 2

𝑥4 × 𝑥0

Try these…

1. 𝑥9 × 𝑥0 2. 4𝑥9 × 𝑥0 3. 5𝑥0 × 𝑥9 4. 5𝑥0 × 4𝑥9

Example 1

4𝑥0

Try these…

1. 3𝑥0 2. −10𝑥0 3. 3

10𝑥0 4. 0.9𝑐0


Exercise

Task 1

1. If 23 × 2𝑎 = 20, what is the value of a?

2. If 4−3 × 4𝑎 = 1, what is the value of a?

3. If 𝑥𝑓 × 𝑥𝑔 = 1 and 𝑓 < 0 and 𝑔 > 0, find possible values for f and g.

Task 2

A cuboid has dimensions as shown.

Show that the volume of the cuboid is 100 cubic metres.

Example 3

𝑥9

𝑥0

Try these…

1. 𝑥5 ÷ 𝑥0 2. 4𝑥5 ÷ 𝑥0 3. 𝑥5 ÷ 8𝑥0 4. 4𝑥5 ÷ 8𝑥0

Example 4

𝑥0 ÷ 𝑥−2

Try these…

1. 𝑥0

𝑥−6 2.

14𝑥0

𝑥−6 3.

𝑥0

7𝑥−6 4.

14𝑥0

7𝑥−6


Everything so far…

Example 1 Example 2 Example 3

Simplify 𝑥9×𝑥3

𝑥4 Simplify

15𝑥9×2𝑥3

10𝑥4 Simplify

24𝑥10

13𝑥5×4𝑥2

Exercise

Task 1

Simplify the following:

1. 𝑥3×𝑥4

𝑥 2.

𝑥2×𝑥6

𝑥3 3.

𝑥7

𝑥4× 𝑥 4.

𝑎−2×𝑎4

𝑎

5. 3𝑠2×2𝑠4

𝑠3 6.

5𝑡4×4𝑡3

2𝑡2 7.

8𝑠9×4𝑠0

2𝑠4×3𝑠−3 8.

2𝑎8

8𝑎3×3𝑎4

9. 15𝑥−4

3𝑥−3×2𝑥−1 10.

16𝑠6×2𝑠5

4𝑠15×3𝑠−4 11.

4𝑎12×5𝑎

72

10𝑎0 12.

6𝑠23×3𝑠

43

3𝑠−2×3𝑠4

Task 2

Fill in the missing exponents:

Task 3

A particle travels 3𝑎𝑏2 metres in 12𝑎2𝑐 seconds.

Calculate the particles average speed in metres per second.


Negative Exponents – Integers

Explore 1 Explore 2

24 = 16 23

27=

2×2×2

2×2×2×2×2×2×2=

1

2×2×2×2=

1

24

÷ 2

23 = 8 Using the exponent rule for division

÷ 2 23

27= 23−7 = 2−4

22 = 4 ∴1

24= 2−4

÷ 2

21 = 2 Can you generalise?

÷ 2

20 = 1

÷ 2

2−1 =

÷ 2

2−2 =

÷ 2

2−3 =

Exercise

For the following terms,

a) Write with a positive exponent b) Evaluate:

1. 26−1 2. 2−1 3. 10−2 4. 2−2

5. −26−1 6. −2−1 7. −10−2 8. −2−2

9. (−26)−1 10. (−2)−1 11. (−10)−2 12. (−2)−2

13. 2−5 12. −7−3 13. (−8)−2 14. (−10)−5

15. (−4)−3 16. 9−4 17. −11−2 18. (−3)−3

19. −3−4 20. 25−2 21. (−2)−6 22. 15−2

Example 1 Example 2

Evaluate 3−2 Evaluate −3−2

Example 3

Evaluate (−3)−2


1. Decide if there are mistakes in the following and explain how to fix the answer:

a) 4−2 = −16 b) 10−3 =1

30

Example 4

Write 1

42 in index form.

Exercise

1. Write in index form:

a) 1

52 b)

1

34 c)

1

83 d)

1

45 e)

1

103 f)

1

26

2. Write in the form 2𝑛:

a) 1

2 b)

1

4 c)

1

8 d)

1

32 e)

1

64 f)

1

256

3. Write in the form 5𝑛:

a) 1

5 b)

1

125 c)

1

625 d)

1

3125 e)

1

78,125 f)

1

390,625

4. Arrange in ascending order:

1

50 5−2

3

10 2−3


Negative Exponents – Fractions

Exercise

Simplify the following:

1. (1

5)−1

2. (2

5)−1

3. (3

5)−1

4. (4

5)−1

5. (1

4)−1

6. (3

4)−1

7. (1

3)−2

8. (2

3)−2

9. (1

5)−2

10. (2

5)−2

11. (3

5)−2

12. (4

5)−2

13. (2

3)−3

14. (4

3)−3

15. (7

8)−2

16. (−1

10)−4

17. (−4

9)−3

18. (−9

10)−2

19. (−5

3)−3

20. (−3

2)−4

21. (−3

10)−3

22. (−3

2)−3

23. (−8

5)−3

24. (−3

8)−2

25. (−6

5)−2

Simplify:

Example 1 Example 2

(3

10)−2

(−3

10)−2


Negative Exponents – algebraic term on numerator

Rewrite the following with positive indices:


𝑥−3 𝑎−2


2𝑥−3 4𝑎−2


1

2𝑥−3

1

4𝑎−2


(2𝑥)−3 (4𝑎)−2

Exercise

Rewrite the following with positive indices:

1. 𝑥−6 2. 𝑥−7 3. 𝑎−8 4. 𝑎−10 5. 𝑝−11

6. 𝑘−14 7. 2𝑠−3 8. 7𝑧−3 9. 3𝑑−4 10. 3𝑑−7

11. 14𝑥−1 12. 12

3𝑥−2 13.

16𝑥−5

4 14.

𝑎−9

2 15.

𝑒−5

6

16. 𝑦−8

3 17.

𝑔−7

10 18.

2𝑥−4

3 19.

4𝑓−5

7 20.

5𝑥−8

9

21. (3𝑥)−3 22. (−3𝑥)−3 23. (5𝑥)−2 24. (−5𝑥)−2 25. (−7𝑎)−3


Negative Exponents – algebraic term on denominator

Exercise

Task 1

Rewrite the following with negative indices:

1. 1

𝑥4 2.

1

𝑥 3.

1

𝑑10 4.

1

𝑏13 5.

1

𝑦𝑥

6. 1

𝑚𝑛 7. 3

𝑥13 8.

5

𝑥9 9.

7

𝑦10 10.

3

𝑥11

11. 4

𝑥3 12.

12

3𝑥2 13.

𝑎

𝑥𝑛 14.

1

2𝑥4 15.

1

3𝑥7

16. 1

8𝑎5 17.

1

7𝑏6 18.

3

4𝑥5 19.

2

5𝑥12 20.

1

8𝑎5

Task 2

Match the equivalent pairs of expressions

(3𝑥)−1 (3𝑥)−2 3𝑥−1 𝑥3 𝑥−3 3𝑥−2

1

𝑥3

1

𝑥−3

1

9𝑥2

3

𝑥

1

3𝑥

3

𝑥2

Rewrite the following with a negative index:


1

𝑥5

1

𝑑10


3

𝑥5

9

𝑑10


1

3𝑥5

9

18𝑑10


Multiplying (2)

Simplify the following then express with a positive exponent:


𝑥9 × 𝑥−11 𝑥7 × 𝑥−14


15𝑎−4 × 2𝑎−6 7𝑎−3 × 8𝑎−5

Task 1

Simplify the following, expressing answers with a positive exponent:

1. 𝑎−2 × 𝑎 2. 𝑥5 × 𝑥−6 3. 𝑥−7 × 𝑥5 4. 𝑥−9 × 𝑥5

5. 𝑎3 × 𝑎−8 6. 𝑓−9 × 𝑓−1 7. 𝑥−4 × 𝑥−2 8. 𝑦−11 × 𝑦−8

9. 𝑏−6 × 𝑏−5 10. 𝑥−3 × 𝑥−4 11. 9𝑥5 × 3𝑥−6 12. 2𝑥−6 × 4𝑥

13. −2𝑎3 × 6𝑎−5 14. −3𝑎4 × −4𝑎−9 15. 1

2𝑥 × 16𝑥−4 16. 2𝑥−1 × 4𝑥−1

17. 3𝑥−2 × 4𝑥−1 18. 5𝑥−4 × 6𝑥−2 19. −2𝑥−6 × −3𝑥−4 20. 2

3𝑥−3 × 12𝑥−4

21. 1

4𝑥−4 × 20𝑥2 22.

3

5𝑦−5 × −

5

3𝑦−3 23.

4

7𝑘−10 ×

5

6𝑘−2 × 21𝑘5

Task 2

Fill in the blanks as many ways as you can:


Dividing (2)

Simplify the following then express with a positive exponent:


𝑦−5

𝑦12

𝑦−6

𝑦14


24𝑐−5 ÷ 6𝑐2 30𝑐−4 ÷ 15𝑐2


3𝑥−8 ÷ 15𝑥−3 4𝑥−9 ÷ 40𝑥−6

Task 1


1. 𝑦−10

𝑦3 2.

8𝑦−10

2𝑦3 3.

𝑦−12

𝑦7 4.

15𝑦−12

3𝑦7

5. 𝑦−10

𝑦−6 6.

3𝑦−10

9𝑦−6 7.

𝑏−13

𝑏−3 8.

2𝑏−13

14𝑏−3

9. 𝑥−8 ÷ 𝑥3 10. 5𝑥−8 ÷ 5𝑥3 11. 𝑥−6 ÷ 𝑥 12. 3

4𝑥−6 ÷

3

4𝑥

13. 𝑥−13 ÷ 𝑥2 14. 16𝑥−13 ÷ 4𝑥2 15. 𝑎−11 ÷ 𝑎−2 16. 24𝑎−11 ÷ 8𝑎−2

17. 𝑥−12 ÷ 𝑥−4 18. 16𝑥−12 ÷ 32𝑥−4 19. 𝑠−14 ÷ 𝑠−6 20. 6𝑠−14 ÷ 9𝑠−6

Task 2



Multiplying (2) & Dividing (2) Combined

Exercise


1. 𝑥2

𝑥3×𝑥4 2.

𝑥5

𝑥×𝑥7 3.

𝑥3×𝑥6

𝑥11×𝑥 4.

𝑥×𝑥6

𝑥2×𝑥3×𝑥4

5. 4𝑥3

2𝑥7×6𝑥8 6.

15𝑥4

3𝑥3×2𝑥 7.

12𝑡8

3𝑡6×6𝑡7 8.

16𝑠6×2𝑠5

4𝑠7×3𝑠8

9. 3𝑎

−32×4𝑎

52

2𝑎−3 10.

5𝑔−15×𝑔2

𝑔−115

11. 4𝑐12×5𝑐

72

10𝑐7 12.

6𝑐23×3𝑐

−43

3𝑐−23×3𝑐

13. 2𝑥6 ×1

4𝑥−2 14. 6𝑥3 ×

1

(2𝑥)−1


Powers of Powers - Numerical

Explore

(22)1 = 22

(22)2 = 22 × 22 = 2 × 2 × 2 × 2 = 24

(22)3 = 22 × 22 × 22 = 2 × 2 × 2 × 2 × 2 × 2 =

(22)4 = 22 × 22 × 22 × 22 =

(22)5 =

(22)𝑛 =

(2𝑚)𝑛 =

Rewrite each expression with a single exponent:

Example Your turn…

(24)3 (34)9

Exercise

Task 1

Rewrite each expression with a single exponent:

1. (32)3 2. (22)4 3. (73)4 4. ((1

3)2)3

5. (0.56)3 6. (49)2 7. (19)9 8. (106)3

9. (73)4 10. ((9

7)1)2

11. (0.93)6 12. ((2

5)3)2


Task 2

Match the expressions which are equivalent.

Complete the blanks to create 6 matching pairs.

23 × 24 212

215 ÷ 23 47 412

46 × 4 48

(42)10 27 4 × 411

Task 3

Look at the statement below:

Three numbers are missing. Write numbers in the boxes to make the statement correct.

Task 4

((−4−3)−2)−1

What does this number mean?

Which order of 1, 2, 3 and 4 makes the highest value?

What about the lowest?

Task 5

Given that 𝑞 = 5𝑛, write 𝑞2 as a single power of 5.


Powers of Powers - Algebraic

Explore

(𝑦3)1 = 𝑦3

(𝑦3)2 = 𝑦3 × 𝑦3 = 𝑦 × 𝑦 × 𝑦 × 𝑦 × 𝑦 × 𝑦 = 𝑦6

(𝑦3)3 = 𝑦3 × 𝑦3 × 𝑦3 = 𝑦 × 𝑦 × 𝑦 × 𝑦 × 𝑦 × 𝑦 × 𝑦 × 𝑦 × 𝑦 =

(𝑦3)4 = 𝑦3 × 𝑦3 × 𝑦3 × 𝑦3 =

(𝑦3)5 =

(𝑦3)𝑛 =

(𝑦𝑚)𝑛 =

Simplify:


(𝑐4)2 (𝑐4)3


−(𝑐4)2 −(𝑐4)3


(−𝑐4)2 (−𝑐4)3


Exercise

Task 1

Simplify:

1. (𝑏5)3 2. −(𝑏5)3 3. (−𝑏5)3 4. (𝑘2)9 5. −(𝑘2)9

6. (−𝑘2)9 7. (𝑝7)10 8. −(𝑝7)10 9. (−𝑝7)10 10. (ℎ3)6

11. −(ℎ3)6 12. (−ℎ3)6 13. (𝑥2)5 14. −(𝑥2)5 15. (−𝑥2)5

16. (𝑗−4)−7 17. −(𝑗−4)−7 18. (−𝑗−4)−7 19. (𝑚−6)−2 20. −(𝑚−6)−2

21. (−𝑚−6)−2 22. (𝑔−9)−4 23. −(𝑔−9)−4 24. (−𝑔−9)−4 25. (𝑎−10)−10

26. −(𝑎−10)−10 27. (−𝑎−10)−10 28. (𝑐−15)−3 29. −(𝑐−15)−3 30. (−𝑐−15)−3

31. (𝑓−4)−9 32. −(𝑓−4)−9 33. (−𝑓−4)−9

Simplify and rewrite with positive exponents:


(𝑐4)−2 (𝑐4)−3


−(𝑐4)−2 −(𝑐4)−3


(−𝑐4)−2 (−𝑐4)−3


Task 2

Simplify and express your answers with positive indices:

1. (𝑏−2)2 2. −(𝑏−2)2 3. (−𝑏−2)2 4. (𝑘−3)4 5. −(𝑘−3)4

6. (−𝑘−3)4 7. (𝑝−5)7 8. −(𝑝−5)7 9. (−𝑝−5)7 10. (ℎ−4)5

11. −(ℎ−4)5 12. (−ℎ−4)5 13. (𝑗2)−3 14. −(𝑗2)−3 15. (−𝑗2)−3

16. (𝑚4)−2 17. −(𝑚4)−2 18. (−𝑚4)−2 19. (𝑔6)−3 20. −(𝑔6)−3

21. (−𝑔6)−3 22. (𝑐7)−2 23. −(𝑐7)−2 24. (−𝑐7)−2

Simplify:

Example 1 Example 2

(3𝑐4)2 (3𝑐−4)2

Example 3 Example 4

(3𝑐4)−2 (3𝑐−4)−2


Exercise

Task 1

Simplify:

1. (2𝑥3)2 2. (2𝑥−3)2 3. (2𝑥3)−2 4. (2𝑥−3)−2

5. (−2𝑥3)2 6. (−2𝑥−3)2 7. (−2𝑥3)−2 8. (−2𝑥−3)−2

9. (5𝑏6)2 10. (5𝑏−6)2 11. (5𝑏6)−2 12. (5𝑏−6)−2

13. (−5𝑏6)2 14. (−5𝑏−6)2 15. (−5𝑏6)−2 16. (−5𝑏−6)−2

17. (10𝑐9)3 18. (10𝑐−9)3 19. (10𝑐9)−3 20. (10𝑐−9)−3

21. (−10𝑐9)3 22. (−10𝑐−9)3 23. (−10𝑐9)−3 24. (−10𝑐−9)−3

25. (5𝑓6)2 26. (5𝑓−6)2 27. (5𝑓6)−2 28. (5𝑓−6)−2

29. (−5𝑓6)2 30. (−5𝑓−6)2 31. (−5𝑓6)−2 32. (−5𝑓−6)−2

33. (10𝑘9)3 34. (10𝑘−9)3 35. (10𝑘9)−3 36. (10𝑘−9)−3

37. (−10𝑘9)3 38. (−10𝑘−9)3 39. (−10𝑘9)−3 40. (−10𝑘−9)−3

41. (−15ℎ9𝑘7)3 42. (3𝑦6)2(𝑥5𝑦2𝑧) 43. (4ℎ3)2(−2𝑔3ℎ)3 44. (14𝑎4𝑏6)2(𝑎6𝑐3)7

Task 2

Simplify:

1. (𝑦4𝑑6)8 2. (−𝑐5ℎ6)4 3. (𝑢4𝑣3)2 4. (𝑥2𝑦2)2

5. (𝑎6𝑐3)7 6. (𝑥𝑦)2(𝑥2𝑦2)2 7. (𝑘9)5(𝑘3)2 8. (3𝑥2𝑦3)2

9. (2𝑘)3(4𝑘3)3 10. (2𝑦2𝑐−3)4 11. (5𝑑𝑐5)3 12. (4𝑟3)2(𝑟2)5

13. (2𝑟−3)2(4𝑟)−3(𝑟3)4 14. (2ℎ3)−3(3ℎ)3 15. (3𝑧2)−2(4𝑧−2)−3

Task 3

Simplify:

1. (𝑥

𝑦)6

2. (5𝑐

𝑑2)2

3. (4𝑑3

𝑐5)3

4. (3𝑤

𝑔6)4

5. (−4𝑠6

𝑡3𝑟5)3

6. (−2𝑑11𝑓6

𝑐18)2

7. (2𝑑4

4𝑒)3

8. (7𝑦2

2𝑥2)2

9. (2𝑥−8

3𝑦11)−2

10. (4𝑐−5

8𝑑0)3

11. (5𝑥13𝑦5𝑧2

3×52)0

12. (3𝑥2

2𝑦2)5

13. (3𝑥

4𝑥2)2

14. (𝑏𝑤

8𝑏2𝑤4)3

15. (4𝑛4𝑏2

7𝑛3𝑏5)2

16. (6𝑤𝑦6

4𝑤4𝑦5)3

17. (9𝑥2

2𝑢2)2

18. (6𝑦2𝑐3

8𝑦𝑐4)2

19. (−3ℎ3

5𝑔5)3

20. (7𝑛2

5𝑛6𝑤5)2


Task 4


Task 5

Solve for y:

(𝑥3)𝑦 =(𝑥𝑦)𝑦

𝑥2

Task 6

1. The statement b a

a bx x is:

A Always True

B Sometimes True

C Never True

2. The statement 1

1x x

is:

A Always True

B Sometimes True

C Never True

3. If 1

1 22 2x then:

A 64x B 4x

C 2x D 16x

4. If b

a a bx x x then:

A a b B 1 b

ab

C 1

ba

b

D

1

ba

b



Task 1

Find the missing exponents:

Task 2

Sam has written the following in a test:

Simplify (5𝑥3𝑦)2

=5x6y2

Is Sam correct? Explain your answer.

Task 3

Fully simplify the following:

1. 𝑏2 × (𝑏5)3 2. 𝑒8 × (𝑒4)−2 3. (𝑚4)−2 ×𝑚4 4. (𝑏3 × 𝑏)6

5. (2𝑐5)2 × 3𝑐3 6. (2𝑚5)3 × (2𝑚3)2 7. (2𝑝)3

8𝑝2 8.

(𝑐2)4

𝑐3

9. (𝑐6)3 ÷ 𝑐2 10. (𝑑−2)4

𝑑−5 11. (5𝑐3)2 ÷ 5𝑐2 12.

18𝑝7

(3𝑝2)2

13. 𝑦8

(𝑦2)3 14.

(𝑏−4)2

𝑏×𝑏3 15. (𝑘3 ÷ 𝑘)6 16.

24ℎ4

(2ℎ32)2

17. (4𝑑

23)2

8𝑑−23

18. (8𝑥𝑦3

𝑥7)−2

3 19. (8𝑥2)0 × (7𝑥0)1 20. ((𝑥−3)2)2

21. (5𝑥3)−1 22. (9𝑥4)1

2 23. (2𝑥)4 ×1

𝑥5 24. 6𝑥3 ×

1

(2𝑥)−1


Expanding Single Brackets Involving Indices

Simplify:


2𝑎3(3𝑎2 + 5𝑎−4) 3𝑎−2(4𝑎5 + 2𝑎)


𝑝1

2 (2𝑝1

2 − 𝑝−3

2) 2𝑝1

3 (3𝑝2

3 − 𝑝−1

3)

Example 3

𝑥2 (𝑥1

3 − 𝑥1

4)

Example 4

𝑛35 (𝑛

12 +

1

𝑛12

)


Task 1

Expand and fully simplify:

1. 𝑥(𝑥3 − 4) 2. 𝑥(𝑥2 − 𝑥−2) 3. 2𝑦−1(3𝑦2 − 2𝑦3)

4. 2𝑦2(3𝑦4 + 5𝑦−2) 5. 𝑚(2𝑚−1 − 4𝑚−4) 6. 3𝑐(4𝑐3 − 6𝑐−4)

7. 4𝑎2(2𝑎−1 + 3𝑎−2) 8. 𝑡−2(3𝑡−2 − 𝑡2) 9. 4𝑑1

2(3𝑑1

2 − 𝑑−1

2)

10. 𝑎−2(𝑎 + 𝑎−1) 11. 2𝑤5 (1

𝑤+ 4𝑤−2) 12. 𝑥2 (𝑥

1

2 + 𝑥1

3)

13. 𝑥1

2(𝑥4 + 𝑥3) 14. 𝑢1

2(3𝑢 + 𝑢3) 15. 𝑥−1

4(𝑥8 + 𝑥6)

16. 3𝑚3

2 (𝑚3

2 +3

𝑚12

) 17. 𝑛1

3 (2𝑛−4

3 −1

𝑛23

) 18. 𝑏1

3(𝑏2 + 2𝑏−1)

19. 𝑥−1

3(𝑥−2 + 𝑥4) 20. 𝑥1

2 (3

𝑥14

−2

𝑥15

) 21. 𝑝3(𝑝−2 + 𝑝3)

22. 𝑥−3(𝑥5 + 𝑥2) 23. 5𝑥1

2 (2𝑥1

2 + 3𝑥3

2) 24. 3𝑎−1(4𝑎3 + 2𝑎)

25. 2𝑢−5(𝑢 + 2𝑢5) 26. 3𝑚2(2𝑚2 + 7𝑚−4) 27. 𝑎1

2 (𝑎1

2 + 𝑎−1

2)

28. 𝑝1

3 (𝑝2

3 + 𝑝−1

3) 29. 𝑒−2

3 (𝑒7

3 − 2𝑒2

3) 30. 5𝑛4 (𝑛−2 +2

𝑛3)

31. 𝑝4 (3𝑝−4 −2

𝑝3) 32. 3𝑎 (𝑎

1

2 + 2𝑎−2) 33. 𝑥1

2(2𝑥 − 3)

34. 2𝑝3

4 (𝑝1

4 − 𝑝) 35. 𝑥2

3 (𝑥1

2 + 𝑥1

4) 36. 𝑥1

5 (2𝑥9

5 +3

𝑥15

)

37. 2𝑎5

3 (1

𝑎23

− 4𝑎4

3) 38. 𝑥−4 (𝑥−1

2 − 𝑥2) 39. 𝑥2 (1

𝑥12

+1

𝑥13

)

40. 𝑥−1

2 (1

𝑥−14

−3

𝑥−16

)

Task 2

Lauren writes down the following statement:

𝑝13 (𝑝

23 − 𝑝−

13) = 𝑝 − 1

Is this statement true? Justify your answer.


Expanding Brackets Involving Indices

Simplify:

Example Your turn…

(2𝑚9 −𝑚−2)(6𝑚−3 +𝑚5) (7𝑥3 − 𝑥−4)(4𝑥−2 + 𝑥9)

Exercise

Expand and fully simplify:

1. (𝑚−2 +𝑚3)(𝑚−2 +𝑚) 2. (𝑥−3 + 𝑥−6)(𝑥7 + 𝑥−2)

3. (𝑥 + 𝑥−1)2 4. (𝑏1

2 + 1)2

5. (3𝑘1

2 − 2)2 6. (𝑥1

2 −1

𝑥12

) (𝑥1

2 +1

𝑥12

)

7. (𝑐1

2 +1

𝑐12

)2

8. (𝑥1

2 +1

𝑥14

) (𝑥1

2 −1

𝑥14

)

9. (𝑥2

3 + 𝑦1

2) (𝑥1

3 − 𝑦1

2) 10. (5𝑐3

4 + 9𝑐1

2) (6𝑐1

3 − 2𝑐1

4)


Unitary Fractional Indices

Explore 1 Explore 2

𝑥1

2 × 𝑥1

2 𝑥1

2 × 𝑥1

2

= (𝑥1

2)2

= 𝑥1

2+1

2

= 𝑥1 = 𝑥1

𝑥1

2 squared is 𝑥 ∴ the square root of 𝑥 is 𝑥1

2 i.e. √𝑥 ∴ √𝑥 = 𝑥1

2

𝑥1

3 × 𝑥1

3 × 𝑥1

3 𝑥1

3 × 𝑥1

3 × 𝑥1

3

= (𝑥1

3)3

= 𝑥1

3+1

3+1

3

= 𝑥1 = 𝑥1

𝑥1

3 cubed is 𝑥 ∴ the cubed root of 𝑥 is 𝑥1

3 i.e. √𝑥3

∴ √𝑥3

= 𝑥1

3

𝑥1

4 × 𝑥1

4 × 𝑥1

4 × 𝑥1

4 𝑥1

4 × 𝑥1

4 × 𝑥1

4 × 𝑥1

4

= (𝑥1

4)4

= 𝑥1

4+1

4+1

4+1

4

= 𝑥1 = 𝑥1

The fourth power of 𝑥1

4 is 𝑥 ∴ the fourth root of 𝑥 is 𝑥1

4 i.e. √𝑥4

∴ √𝑥4

= 𝑥1

4

𝑥1

𝑛 × 𝑥1

𝑛 × 𝑥1

𝑛 × 𝑥1

𝑛 ×⋯ 𝑥1

𝑛 × 𝑥1

𝑛 × 𝑥1

𝑛 × 𝑥1

𝑛 ×⋯

= (𝑥1

𝑛)𝑛

= 𝑥1

𝑛+1

𝑛+1

𝑛+1

𝑛+⋯

= 𝑥1 = 𝑥1

The nth power 𝑥1

𝑛 is 𝑥 ∴ the nth root of 𝑥 is 𝑥1

𝑛 i.e. √𝑥𝑛

∴ √𝑥𝑛

= 𝑥1

𝑛


Task 1

Rewrite the following using the radical sign:

1. 𝑥1

5 2. 𝑥1

6 3. 𝑥1

7 4. 𝑥1

8 5. 𝑥1

9 6. 𝑥1

10 7. 𝑥1

𝑚

8. 𝑥−1

2 9. 𝑥−1

3 10. 𝑥−1

4 11. 𝑥−1

5 12. 𝑥−1

6 13. 𝑥−1

7 14. 𝑥−1

𝑎

Task 2

Rewrite the following using the radical sign:

1. 4𝑥1

3 2. 4𝑥−1

3 3. 1

4𝑥1

3 4. 1

4𝑥−

1

3

5. 15𝑦1

10 6. 15𝑦−1

10 7. 1

15𝑦1

10 8. 1

15𝑦−

1

10

9. −14𝑥1

8 10. −14𝑥−1

8 11. −1

14𝑥1

8 12. −1

14𝑥−

1

8

13. 18𝑐1

2 14. 18𝑐−1

2 15. 1

18𝑐1

2 16. 1

18𝑐−

1

2

17. −20𝑘1

9 18. −20𝑘−1

9 19. −𝑘19

20 20. −

𝑘−19

20

21. 3𝑐1

7 22. 3𝑐−1

7 23. 𝑐17

3 24.

1

3𝑐−

1

7

Task 3

Rewrite the following using fractional exponents:

1. √𝑎9

2. √𝑘4

3. √𝑝5 4. √𝑥

8 5. √𝑚

3

6. √𝑧7

7. √𝑤10

8. √𝑡14

9. 1

√𝑎9 10.

1

√𝑘4

11. 1

√𝑙10 12.

1

√ℎ7 13.

1

√𝑝13 14.

1

√𝑚7 15.

1

√𝑣16


Evaluate:


641

2 641

3


64−1

2 64−1

3


(81

16)

1

4 (

81

16)

1

2


(81

16)−1

4 (

81

16)−1

2


Task 1

Evaluate

1. 251

2 2. 25−1

2 3. 161

2 4. 16−1

2

5. 91

2 6. 9−1

2 7. 1001

2 8. 100−1

2

9. 361

2 10. 36−1

2 11. 81

3 12. 8−1

3

13. 1251

3 14. 125−1

3 15. 10001

3 16. 1000−1

3

17. 161

4 18. 16−1

4 19. 321

5 20. 32−1

5

21. 2161

3 22. 216−1

3 23. 5121

3 24. 512−1

3

25. (25

49)

1

2 26. (

25

49)−1

2 27. (

36

121)

1

2 28. (

36

121)−1

2

29. (625

1296)

1

4 30. (

625

1296)−1

4 31. (

216

343)

1

3 32. (

216

343)−1

3

33. (343

1000)

1

3 34. (

343

1000)−1

3 35. (

125

729)

1

3 36. (

125

729)−1

3

Task 2

a) Given that 𝑦 = 𝑥1

3, find y when 𝑥 = 64.

b) Given that 𝑦 = 𝑥1


c) Given that 𝑦 = 𝑥1


d) Given that 𝑦 = 𝑥1


Task 3

For 𝑎 > 1, put the following in order of size from smallest to largest:

𝑎0, 𝑎2, 𝑎, 𝑎−2, 𝑎12


Task 4

Solve for a:

𝑥 × 𝑥4 × 𝑥𝑎

√𝑥= √

(𝑥𝑎)8

(1𝑥4)

Task 5

The statement 𝑥 > 𝑥1

2 is:

Always True

Sometimes True

Never True


Non-Unitary Fractional Indices

Explore – Numeric Explore – algebraic

81

3 = √83

= 2 𝑥1

5 = √𝑥5

82

3 = (81

3)2

= (√83)2= (2)2 = 4 𝑥

2

5 = (𝑥1

5)2

= (√𝑥5)2

83

3 = (81

3)3

= (√83)3= (2)3 = 8 𝑥

3

5 = (𝑥1

5)3

= (√𝑥5)3

84

3 = (81

3)4

= (√83)4= (2)4 = 16 𝑥

4

5 = (𝑥1

5)4

= (√𝑥5)4

85

3 = (81

3)5

= (√83)5= (2)5 = 32 𝑥

𝑚

5 = (𝑥1

5)𝑚

= (√𝑥5)𝑚

8𝑚

3 = (81

3)𝑚

= (√83)𝑚= (2)𝑚 𝑥

𝑚

𝑛 = (𝑥1

𝑛)𝑚

= (√𝑥𝑛)𝑚

Task 1

Write in index form:

1. √𝑎3 2. 5√𝑎3 3. 1

√𝑎3 4.

5

√𝑎3

5. √𝑤5 6. 4√𝑤5 7. 1

√𝑤5 8.

4

√𝑤5

9. √𝑥23

10. 9√𝑥23

11. 1

√𝑥23 12.

9

√𝑥23

13. √𝑤43

14. -6√𝑤43

15. 1

√𝑤43 16. −

6

√𝑤43

17. √𝑚25

18. −2√𝑚25

19. 1

√𝑚25 20. −

2

√𝑚25

21. √𝑘49

22. −12√𝑘49

23. 1

√𝑘49 24. −

12

√𝑘49

25. √𝑥5𝑦76

26. √𝑎7𝑏3𝑐4

27. √𝑝2𝑞3𝑟47

28. √𝑝2𝑞3𝑟417


Task 3

Find equivalent terms from the following:

Task 2

Write with a radical sign in simplest form:

1. 𝑥3

4 2. 𝑚7

5 3. 𝑝3

2 4. 𝑦5

8 5. 𝑐4

3

6. 4 (𝑥3

4) 7. 9 (𝑚7

5) 8. 16(𝑝3

2) 9. 5 (𝑦5

8) 10. 10(𝑐4

3)

11. 𝑥−3

4 12. 𝑚−7

5 13. 𝑝−3

2 14. 𝑦−5

8 15. 𝑐−4

3

16. 4 (𝑥−3

4) 17. 9 (𝑚−7

5) 18. 16(𝑝−3

2) 19. 5 (𝑦−5

8) 20. 10(𝑐−4

3)

Evaluate:


253

2 813

4


25−3

2 81−3

4


Task 1

Evaluate:

1. 84

3 2. 275

3 3. 645

6 4. 813

2 5. 6253

4

6. 93

2 7. 1003

2 8. 31252

5 9. 85

3 10. 45

2

11. 813

4 12. 125−2

3 13. 243−2

5 14. 32−2

5 15. 16−3

4

16. 4−5

2 17. 64−2

3 18. 8−4

3 19. 32−4

5 20. 125−2

3

21. 1000−2

3 22. 81−3

4 23. 625−3

4 24. (−8)4

3 25. (−27)5

3

26. (−3125)2

5 27. (−8)5

3 28. (−125)−2

3 29. (−243)−2

5 30. (−32)−2

5

31. (−64)−2

3 32. (−8)−4

3 33. (−32)−4

5 34. (−125)−2

3 35. (−1000)−2

3

36. (4

25)

3

2 37. (

4

25)−3

2 38. (

8

125)

2

3 39. (

8

125)−2

3 40. (

49

100)

3

2

41. (49

100)−3

2 42. (

8

125)

2

3 43. (

8

125)−2

3 44. (

16

100)

5

2 45. (

16

100)−5

2

46. (64

1000)

4

3 47. (

64

1000)−4

3 48. (

16

25)

3

2 49. (

16

25)−3

2 50. (

32

243)

6

5

Task 2

a) Arrange the following in ascending order: b) Which is the odd one out? Explain your answer.

251

2, 82

3, 271

3 641

2, 163

4, 92

3, 43

2

Task 3

Gina has completed her homework.

Can you spot any mistakes?

Task 4

a) Given that 𝑦 = 𝑥3

5, find 𝑦 when 𝑥 = 32. b) Given that 𝑦 = 𝑥3

2, find 𝑦 when 𝑥 = 49.

c) Given that 𝑦 = 𝑥2

3, find 𝑦 when 𝑥 = 1000. d) Given that 𝑦 = 𝑥3

4, find 𝑦 when 𝑥 = 81.

e) Given that 𝑦 = 𝑥5

6, find 𝑦 when 𝑥 = 64. f) Given that 𝑦 = 𝑥3

4, find 𝑦 when 𝑥 = 10,000.


Task 5

Find equivalent pairs from the following:

Task 6

1. (4𝑥)3

4 2. (9𝑚)7

5 3. (16𝑝)3

2 4. (5𝑦)5

8 5. (10𝑐)4

3

6. (4𝑥)−3

4 7. (9𝑚)−7

5 8. (16𝑝)−3

2 9. (5𝑦)−5

8 10. (10𝑐)−4

3

11. (4𝑥6

9𝑦4)

5

2 12. (

243𝑥5

32𝑦20)

3

5 13. (

64𝑥3

27𝑦6)

2

3


Task 7

Match up each expression from the first column with its partner from the second column, and

write the answers in the table below

First column Second column 1 x 11

2 3

1 1

x x

A 32x

K 12x

2 1

x

12 2

3 x B 31

2x

L 92x

3 1

x

13 7x C 11

2x

M 2x

4 3 x 14

8

1

x

D 4x

N 32x

5 2

1

x

15 8

1

x

E 1x O 12x

6 3

1

2x

16 6x x x F 1

2x P 1

2x

7 3

2

x

17

2

4

x

G 4x Q 5x

8 x x 18

2

1

4x

H 2x R 2

3x

9 2

x

x

19 2

4

x

I 2x S 32x

10 2

1

x

20 24x J 1

3x T 72x

Table for Answers:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20



Task 1

Match up each expression from the first column with its partner from the second column and write

the answers in the table below.

First column Second column 1 x 11

2 3

1 1

x x

A 32x

K 12x

2 1

x

12 2

3 x B 31

2x

L 92x

3 1

x

13 7x C 11

2x

M 2x

4 3 x 14

8

1

x

D 4x

N 32x

5 2

1

x

15 8

1

x

E 1x O 12x

6 3

1

2x

16 6x x x F 1

2x P 1

2x

7 3

2

x

17

2

4

x

G 4x Q 5x

8 x x 18

2

1

4x

H 2x R 2

3x

9 2

x

x

19 2

4

x

I 2x S 32x

10 2

1

x

20 24x J 1

3x T 72x

Table for Answers:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20


Task 2

Simplify the following, expressing answers with a radical sign where appropriate:

1. 𝑚4 × √𝑚 2. 𝑎−3 × √𝑎 3. 𝑢2 × √𝑢23

4. √𝑐54

× 𝑐 5. 𝑒3 ÷ √𝑒 6. 𝑚−4 ÷ √𝑚

7. √𝑘23

÷ 𝑘 8. 𝑓−2 ÷ √𝑓4 9.

√𝑏

√𝑏3

10. 4𝑝−2 ÷ 2√𝑝3 11. √𝑎 ×

1

𝑎−2 12. √𝑎 × 𝑎2

13. 1

𝑎−3× √𝑎

3× √𝑎3

4 14. √𝑝3

4× √𝑝5

3 15. √𝑥 × √𝑥2

3

16. 3𝑚 × √𝑚3

17. 4𝑝 × √𝑝23

18. 5𝑥2 × 3𝑥1

2

19. 3𝑥2 × 7𝑥1

3 20. 𝑥3 × 2𝑥1

2 21. 9𝑥2 × 3𝑥−1

2

22. 3𝑥3 × 10𝑥−1

3 × 4𝑥2 × 3𝑥−1

2 23. 4√𝑎3

3𝑎

24. The area of a circle is 𝑥3.

Find an expression for the radius of the circle (express answers both in exponent form and with a

radical sign).

What if 𝑥3 was the area of a semi-circle?

Quarter-circle?


Task 3


Solving Exponential Equations

Solve:


𝑥3

4 = 27 𝑥2

3 = 16


𝑥−3

4 = 27 𝑥−2

3 = 16

Task

Solve:

1. 𝑥1

5 = 2 2. 𝑥1

4 = 3 3. 𝑥1

3 = 3 4. 𝑥2

3 = 9 5. 𝑥4

3 = 16

6. 𝑥5

3 = 243 7. 𝑥2

3 =1

4 8. 𝑦−3 = 3

3

8 9. 𝑥−

1

2 =1

7 10. 𝑥−

3

2 =8

27

11. 𝑥−5

2 =1

32 12. 𝑥−

2

3 = 27

9 13. 𝑥−

2

3 = 71

9

14. If 𝑥1

2 = 6 and 𝑦−3 = 64, find 𝑥

𝑦.

15. If 𝑥3

2 = 8 and 𝑦−2 =25

4, find

𝑥

𝑦.


Changing Bases

What do you notice about 2, 8, 4 and 64 for example?

What about 9, 27, 81…?

Find the value of 𝑥:

Example 1 Example 2

3𝑥 = 27 5𝑥 =1

25

3𝑥 = 33 5𝑥 =1

52

𝑥 = 3 5𝑥 =

𝑥 =

Example 3 Example 4

9𝑥 =1

3 9𝑥 = √27

(32)𝑥 = 3−1 (32)𝑥 = √33

32𝑥 = (32)𝑥 = 3⬚

2𝑥 = 32𝑥 =

𝑥 = 2𝑥 =

𝑥 =


Task 1

1. Express each of the following in the form 7𝑘:

a) √74

b) 1

7√7 c) 7 × 4910

2. Solve:

1. 4𝑥 = 64 2. 5𝑥 = 25 3. 2𝑥 = 4 4. 4𝑥 = 32

5. 9𝑥 = 27 6. 16𝑥 = 32 7. 3𝑥 = 1 8. 4𝑥 = 1

9. 25𝑥 = 125 10. 4𝑥 = 128 11. 100𝑥 = 1000 12. 5𝑥 =1

125

13. 2𝑥 =1

4 14. 3𝑥 =

1

3 15. 16𝑥 = 4 16. 8𝑥 = 2

17. 27𝑥 = 3 18. 16𝑥 = 4 19. 125𝑥 =1

5 20. 8𝑥 =

1

2

21. 25𝑥 =1

5 22. 4𝑥 = √32 23. 3𝑥 = √27 24. 8𝑥 = √32

25. 4𝑥 = √8 26. 23𝑥 = √64 27. 25𝑥 = √64 28. 8𝑥 = √128

29. 4𝑥 = √323

30. 9𝑥 = √275

31. 625𝑥 = √1254

32. 8𝑥 = √324

33. 9𝑥 = √813

34. 1000𝑥 = √1005

3. a) Express 1

16 as a power of 2.

b) Express √8 as a power of 2.

c) Express 2 as a power of 8.

d) Express 520 as a power of 25.

4. a) Given 32√2 = 2𝑎 , find 𝑎.

b) If 9√3 = 3𝑘 , find 𝑘.

c) Solve for 𝑥:

3 × 5𝑥 =3

125

5. Express in simplest exponent form then evaluate:

a) 641

3 × 23 b) 272

3 ÷ 93

2

6. a) Show that 93

2 = 27.

b) Hence, or otherwise, solve the equation 9𝑥 = 274

7. Show that 81

3 × 2−5 = 4−2


8. Solve:

a) 42𝑥 + 1 = 65 b) 32𝑥−1 = 27 c) 53𝑥−8 = 252𝑥

d) 32𝑥−1 = 27𝑥 e) 3𝑥 = 92𝑥−1 f) 42𝑥+1 = 82𝑥−1

9. a) 8𝑥 = 29 b) 5𝑥 = 5√5 c) 4𝑥 =86

24

d) 27𝑥 = √930 e) √93× √27

4= √3

𝑥

10. a) 161

5 × 2𝑦 = 83

4. Solve for 𝑦.

b) 271

4 × 3𝑦 = 92

3. Solve for 𝑦.

c) 251

3 × 125𝑦 = 54

5. Solve for 𝑦.

d) 7−𝑛 = 0.5. Find the value of (75)𝑛.

e) 5−𝑛 = 0.1. Find the value of (53)𝑛.

f) 3−𝑛 = 0.2. Find the value of (34)𝑛.

g) Express 82𝑥+3 in the form 2𝑦, stating 𝑦 in terms of 𝑥.

11. a) If 4𝑥 + 4𝑥 + 4𝑥 + 4𝑥 = 416, find the value of 𝑥.

b) If 9𝑥 + 9𝑥 + 9𝑥 = 311, what is the value of 𝑥?

c) If 2𝑥 + 2𝑥 = 219, what is the value of 𝑥?

d) Solve 2𝑥 = √2 + √2 + √2 + √2

12. a) If 𝑥 = 3𝑎 and √9𝑥 = 3𝑏 , express 𝑎 in terms of 𝑏.

b) Given that 𝑥 = 2𝑘 and √4

𝑥= 2𝑐 , find 𝑐 in terms of 𝑘.

c) 27−𝑥

3 + 811−𝑥

4 . Express in the form 𝑎

𝑏𝑥.

d) Given that (√𝑥𝑏)𝑎 = 𝑥𝑎 × 𝑥𝑏 , find an expression for 𝑎 in terms of 𝑏.

e) Solve these simultaneous equations:

(𝑥𝑎)2 = (𝑥4)2𝑏 and (√𝑥)𝑎 =𝑥3

𝑥𝑏

13. 𝑥 = 2𝑝 and 𝑦 = 2𝑞

a) Express in terms of 𝑥 and/or 𝑦:

(i) 2𝑝+𝑞 (ii) 22𝑞 (iii) 2𝑝−1

b) Given that 𝑥𝑦 = 32 and 2𝑥𝑦2 = 32, find the values of 𝑝 and 𝑞.


14. 𝑥 = 5𝑚 and 𝑦 = 5𝑛

a) Write 5𝑚−𝑛 in terms of 𝑥 and 𝑦.

b) Write 53𝑛 in terms of 𝑦.

c) Write 5𝑚+2 in terms of 𝑥.

d) Write 5𝑚+𝑛

2 in terms of 𝑥 and 𝑦.

15. Let us denote 3100 by 𝑀. Express each of the following in terms of 𝑀:

a) 3101 b) 3100 − 2 × 3101 + 3102 c) 399 d) 9100

16. A formula is given as 𝑃 =27

3𝑥

a) Calculate P when 𝑥 = 2.

b) Calculate 𝑥 when 𝑃 = 9.

c) Calculate the maximum value of P given 𝑥 ≥ 0.

17. The intensity of light, I, emerging after passing through a liquid with concentration, c, is given by the

formula

𝐼 =20

2𝑐 where 𝑐 ≥ 0

a) Find the intensity of light when the concentration is 3.

b) Find the concentration of the liquid when the intensity is 10.

c) What is the maximum possible intensity?


Final Task

Evaluate the following:

a) 25 b) 2−3 c) (22)−3 d) 23

25

e) 24×34

64 f) (

2

32)−2

× (22

33) g) (

5

2)10

((5

4)5

)−2

h) 5×(32×10)2

32×602

i) 3 × ((2 × 3)−1 ×1

23)−1

× (3 × 22)−2 j) (1 − 2 × (24

5)−1

×23

5)100

k) (32)3 × (2 × 35)−2 × 182 l) ((2

52)2

(63

22× (

2

3)−1

)−2

)

−1

m)

(

(2×393)

−2

(9

4)2×(2

5)−1

)

−1

n) 𝑎2(23×𝑐−2)

((𝑎

2)3)−2 − 2(

𝑐

(𝑎2×2−1)2)−2

o) (23

𝑎2÷𝑐3

32)−2

((𝑐3

𝑎2)−1

÷(1

𝑏−1))

−1

indices...exercise task 1 1. 3if 2×2 Ô=20, what is the value of a? 2. if 4−3×4 Ô=1, what is...

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