Chapter 2 2tb2 Print

NAME: ………………………………………………………. CLASS: 2TB2 DATE: ………………………

1

CHAPTER 2: SQUARES, SQURE ROOTS, CUBES AND CUBE ROOTS

To square a number means to multiply the number by itself

Stating a number multiplied by itself as a number to the power of two and vice versa.

Write the following as a number to the power of two

1. 9 × 9 = 92

2.

𝟓

𝟕×

𝟓

𝟕= 3. −2

2

5× (−2

2

5) = 4. 1.55 × 1.55 =

5. −11 × (−11) =

6. −𝟓

𝟕 × −

𝟓

𝟕 =

7. −2.1 × (−2.1) =

8. −9 × (−9) =

9. 0.5 × 0.5 =

10. 12

3× 1

2

3=

Expand each of the following as a number multiplied by itself.

1. −7 2 = −7 × (−7)

2. 12

5

2=

3. –0.75 2

=

4. −1

2

3

2

5. 2

5

2=

6. 82 = 7. 0.782 = 8. 582 =

Determining the squares of numbers without using a calculator.

1. 12 = 1 × 1 = 1

2. 22 = 3. 32 = 4. 42 = 5. 52 =

6. 62 = 6 × 6 =………………

7. 72 = 8. 82 = 9. 92 = 10. 102 =

11. −11 2 = −11 × (−11) = 121

12. −12 2 13. −13 2 14. −14 2 15. −15 2

16. 1.62 = 1.6 × 1.6 = 2.56

17. 1.72 = 18. 1.82 = 19. 1.92 = 20. 2.12 =

21. −2.2 2 = −2.2 × (−2.2)

= 4.84

22. −0.23 2 23. −0.24 2 24. −0.25 2 25. −2.6 2

26. 2

5

2=

2

5×

2

5

=4

25

27. 1

4

2= 28.

5

9

2= 29.

1

4

2= 30.

11

12

2=

31. −2

5

2

= −2

5× (−

2

5)

=4

25

32. −2

3

2

33. −2

5

2

34. −2

15

2

35. −6

7

2


2

36. 12

5

2

= 7

5

2

=7

5×

7

5

=49

25

= 124

25

37. 12

3

2

38. 11

2

2

39. 12

7

2

40. 22

3

2

41. −12

3

2

= −5

3

2

= −5

3 × −

5

3

=25

9

= …………..

42. −21

2

2

43. −32

4

2

44. −41

2

2

45. −22

3

2

Estimating the squares of numbers.

≈ means “ is approximately equal to”

1. 2912 2912 ≈ 3002 2912 ≈ 90 000

2. 422 3. 0.782

4. −0.51 2

5. −12.3 2

6. 5.92

7. 1792

8. −0.019 2

9. 69.12

Determining the squares of numbers using a calculator

1. −8.8 2 =

2. 122

13

2=

3. –0.175 2

=

4. −1

2

3

2

5. 13

15

2=

6. 8982 = 7. 10.782 = 8. 5.182 =

Perfect squares

A non-zero whole number that can be expressed as a number multiplied by itself is a ……………………………………………….

1. List the perfect square between 1 and 100.

…………………………………………………………………………………………………………………………………………………………………………

2. List the perfect square less than 20

………………………………………………………………………………………………………………………


3

Determining if a number is a perfect square (prime factorisation method)

A non-zero whole number that can be expressed as a number multiplied by itself is a perfect square.

1. 324 324 = (2 x 3 x 3) x (2 x 3 x 3) = 18 x 18 = 182

324 is a perfect square

2. 168 168 is …………………………………………….

3. 50 50 is …………………………………………….

4. 144

144 is …………………………………………….

5. 196 196 is …………………………………………….

6. 216 216 is …………………………………………….

7. 256 256 is ……………………………………………….

8. 348 348 is …………………………………………….

9. 361 361 is …………………………………………….

Square Roots of Positive Numbers

is the symbol for square root.

since 𝑎 × 𝑎 = 𝑎2, then 𝑎2 = 𝑎 × 𝑎 = 𝑎

Stating the square root of a positive number as the number multiplied by itself equal to the given number

Given that 52 = 25, find 25

25 = 52 = …………

Given that 62 = 36, find 36 Given that 72 = 49, find 49

Given that 2

3

2 =

4

9, find

4

9

4

9 =

2

3

2

= ………………

Given that 3

4

2 =

9

16, find

9

16

Given that 3

4

2 =

9

16, find

9

16


4

Given that 0.22 = 0.04 , find 0.04

0.04 = 0.22 = ………….

Given that 0.072 = 0.0049 , find

0.0049

Given that 1.22 = 1.44 , find 1.44

Determining the square roots of perfect squares without using a calculator.

144

= (2 × 2 × 3) × (2 × 2 × 3)

= 12 × 12

= 122 = …………………..

256 256

225 400 900

625 169 576

Determining the square roots of positive numbers without using a calculator.

25

81 =

21

4 1

7

9

25

49 =

196

225

18

50


5

0.36 1.44

4.41

Multiplying two square roots

a × b = ab

5 × 5 = 52 = ………………..

3

5×

3

5=

0.49 × 0.49 =

11 × 11 = 1

2

7× 1

2

7=

3.21 × 3.21 =

25 × 64 = 1600 = …………….

1

4× 1

7

9 =

1

4×

16

9

= 16

36

= 4

6

= …………….

0.5 × 32 = 16 = ………….

8 × 32 =

2 × 11

8=

1.5 × 24 =

6 × 24 =

3

8× 1

1

2=

0.3 × 1.2 =

2 × 18 =

5

6× 1

7

8=

3 × 0.75 =


6

Estimating square roots of positive numbers


65 ≈ 64

65 ≈ …………….

8.9 15

168

9.2 170

224

15.9 288.7

Finding the square roots of positive numbers using a calculator

Use a scientific calculator to find the following, correct to 3 decimal places.

68 132.7

2

7

32

5

12.712 2

3

4

Posing and solving problems involving squares and square roots

A square carpet with side 4 m is placed on a square floor of area 81 m2. What area of the floor is not covered by the carpet? Answer: 65 m2

The figure shows a rectangular book next to a square book. The area of the square book is 256 cm2. If the length of the rectangular book is 20 cm, find the area of the rectangular book. Answer: 320 cm2

Given a right-angled triangle with the

height 5 cm and the base 45

5 cm.

Find the area for the triangle. Answer: 1.5 cm2


7

Given that 0.3252 = 0.1056, find the value of 32.52. 32.52= 0.3252 x 1002

= 0.1056 x 10000

= ……………….

Given that 1.122 = 1.254, find the value of 11.22. 11.22 = …………….…. x …….…………

= ……………….. x ………….…….

= ……………….….

Given that 2.3212 = 5.387, find the value of 232.12. 232.12 = …………….…. x …….…………

= ……………….. x ………….…….

= ……………….….

Given that 0.5292 = 0.2798, find the value of 52.92.



Given that 3.912 = 15.29, find the value of 0.3912. 0.3912 = 3.912 ÷ 102

= ……………….. ÷………………

= ……………….


= …………..…… ÷ …………..…

= ……………….


= …………..…… ÷ …………..…

= ……………….




Given that 5.92 = 34.81, find the value of 5902.




8

Given that 2.5 = 1.581, then find

the value of 250.

250 = 2.5 × 100

= 1.581 x 10

= …………………


the value of 360.


the value of 490.


the value of 810.

Given that 10 = 3.162, then find

the value of 1000.

Given that 2 = 1.414, then find the

value of 200.


the value of 0.081.

0.081 = 8.1 ÷ 100

= 2.846 ÷ 10

= ……………………..


the value of 0.1.

0.1 = 10 ÷ 100

= …………. ÷ ……………..

= ……………………..


value of 0.02.

0.02 = 2 ÷ 100

= …………. ÷ ……………..

= ……………………..


the value of 0.025.


the value of 0.00036.






the value of 0.001.


value of 20000.


9

Given that 1.1452 = 1.3, then 130.

1.3 = 1.145

130 = 1.3 × 100

= 1.145 x 10

= ………………………

Given that 2.582 = 6.67, then 667.

6.67 = 2.58

667 = 6.67 × 100

= ……………. x …………….

= ………………………

Given that 3.332 = 11.09, then 1109.

11.09 = 3.33

1109 = 11.09 × 100

= ……………. x …………….

= ………………………

Given that 3.1152 = 9.703,

then 97030.

Given that 1.1232 = 1.262,

then 126.2.

Given that 0.2122 = 0.045, then 4.5.

Given that 3.1282 = 9.784,

then 0.09784.

9.784 = 3.128

0.09784 = 9.784 ÷ 100

= 3.128 ÷ 10

= ………………………

Given that 1.092 = 1.188,

then 0.01188.

1.188 = 1.09

0.01188 = 1.188 ÷ 100

= ……….…….. ÷ …………….

= ………………………

Given that 25.632 = 656.8,

then 0.06568.

656.8 = 25.63

0.06568 = 656.8 ÷ 10000

= ……….…….. ÷ …………….

= ………………………

Given that 1.1282 = 1.272,

then 0.01272.

Given that 3.082 = 9.4864,

then 0.094864.

Given that 88.882 = 7899.7,

then 78.997.

Given that 1.252 = 1.5625,

then 0.015625.

Given that 13.082 = 171.09,

then 1.7109.

Given that 8.882 = 78.85,

then 0.7885.


10


95.02 = 0.9502 x 1002

= 0.9025 x 10000

= ……………………

Given that 2.1452 = 4.6, then 46000.

4.6 = 2.145

46000 = 4.6 × 10000 = 2.145 x 100

= …………….

1

9× 2

1

4

2

1

9× 2

1

4

2

= 1

3×

9

4

2

= 1

3×

3

2

2

= 1

2

2

= ……………


Given that 2.3242 = 5.4, then 54000

54000 =

1

36× 2

1

4

2

Find the value of 16

16 = 4 = ………

Given that 𝑥 = 5, 𝑦 = 4, calculate

the value x – y

𝑥 = 5

𝑥 2

= 52

𝑥 = …….

𝑦 = 4

𝑦 2

= 42

𝑦 = …….

𝑥 – 𝑦 = ………….. −……………. = ……………

Given that 8.1 = 2.846, find the

value of 81 + 0.081

Find the value of 81

Given that 𝑥 = 4, 𝑦 = 3, calculate

the value x – y


value of 810 + 0.081


11

Cubes of Number

To cube a number is to multiply the number by itself twice

Stating a number multiplied by itself twice as a number to the power of three and vice versa

Write the following number as a number to the power of three

9 × 9 × 9 =

𝟓

𝟕×

𝟓

𝟕 ×

𝟓

𝟕= −2

2

5× (−2

2

5) × (−2

2

5) = 1.55 × 1.55 × 1.55 =

−𝟓

𝟕 × −

𝟓

𝟕 × −

𝟓

𝟕 =

−2.1 × (−2.1) × (−2.1) =

−9 × (−9) × (−9) =

0.5 × 0.5 × 0.5 =

Write each of the following as a number multiplied by itself twice.

1. −7 3 = −7 × −7 × (−7)

2. 12

5

3=

3. –0.5 3

=

4. −1

1

2

3

5. 2

5

3=

6. 6 3 = 7. 0.283 = 8. 113 =

Determining the cubes of numbers without using a calculator

1. 13 = 1 × 1 × 1 = 1 × 1 = 1

2. 23 = 3. 33 = 4. 43 =

5. 53 = 5 × 5 × 5 = 25 × 5

= ………………..

6. 63 = 7. 73 = 8. 83 =

9. 93 =

10. 103 = 11. (−1)3 = 12. (−2)3 =

13. (−3)3 =

14. (−4)3 =

15. (−5)3 =

16. (−6)3 =

17. (−7)3 =

18. (−8)3 =

19. (−9)3 =

20. (−10)3 =


12

21. 3

5

3=

22. 2

7

3=

23. 1

2

3=

24. 5

6

3=

25. −1

5

3=

26. −3

4

3=

27. −3

7

3=

28. −5

9

3=

29. −13

4

3=

= −7

4

3

= −7

4× −

7

4 × −

7

4

=49

16× −

7

4

= −343

64

= −523

64

30. −21

4

3=

31. −13

5

3=

32. −11

3

3=

33. 21

4

3

= 9

4

3

= 9

4×

9

4×

9

4

= 729

16

= …………….

34. 11

4

3=

35. 31

2

3=

36. 21

3

3=

37. 0.13 = 0.1 × 0.1 × 0.1

= 0.01 × 0.1 = ………………

38. 0.23 39. 0.033 40. 0.043


13

41. −1.1 3 42. −0.7 3 43. −0.08 3 44. −0.07 3

Estimating cubes of numbers


0.83 0.83 ≈ 13 0.83 ≈ 1

4.23 3.123

(−1.99)3

(−71.8)3 (−49.1)3

(−9.9)3 3.93 18.43

Determining cubes of numbers using a calculator

553

−773 0.993 −6.93

11

13

3

12

3

3

−21

5

3

−8

9

3

Cube Roots of Number

𝟑

is the symbol for the cube root of a number.

since 𝑎 × 𝑎 × 𝑎 = 𝑎3, then 𝑎33= 𝑎

Stating the cube root of a number as the number multiplied by itself twice equals to the given number.


1253

= 533

= …………..




14

Given that (−2)3 = -8, find −83

Given that (−0.3)3 = −0.027, find

−0.0273

Given that (−0.9)3 = −0.729, find

−0.7293

Given that 2

3

3=

8

27, find

8

27

3 .

Given that −2

5

3= −

8

125 , find

−8

125

3 .

Given that −3

4

3= −

27

64 , find

−27

64

3 .

Determining the cube roots of integers without using a calculator

1253

= 5 × 5 × 53

= ……………….

−2163

−83

17283

−5123

7293

Determining the cube roots of numbers without using a calculator

8

125

3

= 2

5×

2

5×

2

5

3

= 2

5

33

= …………

1

8

3

64

343

3


15

−64

343

3

= −4

7× −

4

7 × −

4

7

3

= −4

7

33

= …………………

−1

125

3

−8

27

3

−417

27

3

= −125

27

3

= −5

3× −

5

3 × −

5

3

3

= −5

3

33

= −5

3

= ………………

33

8

3

−191

125

3

−0.2163

= −0.6 × −0.6 × (−0.6) 3

= (−0.6)3 3

= ………….

−0.0643

−0.0013

Estimating cube roots of numbers


−73

≈ −83

−73

≈ …………….

−263

713

2123

126.93

7.43

283

−1393

0.2313


16

Determining cube roots of numbers using a calculator

Use your scientific calculator to evaluate each of the following, correct to 4 decimal places where necessary.

383

−69.43

−763

18.33

2

7

3

12

3

3

Posing and solving problems involving cubes and cube roots.

The total surface area of a cube-shaped box is 384 cm2. What is its volume?

6x

2 = …………..

x2 = …………. ÷ ………..

x2 = …………..

x = ……

= ………….

The volume of the cube

= …………. x …………..x ………….

= …………… cm3

A cube with length 27 cm being cut into 27 smaller cubes. Find the total surface area of the small cubes and the volume for a small cube.

The volume of a small cube = (………. x …………x ………….) ÷ 27

= ……………..cm3

The length of each side of a small cube

= ……… .

3

= …………. cm The total surface area of the small cubes = 27 x ………. x …………. x ……………

=………………….. cm2

A cubic tank is completely filled with water. Each minute, 0.25 m3 of water leaked out through a hole at the bottom of the tank. If it takes 108 minutes for the tank to be completely drained, what is the length of each side of the tank? Answer: 3 m

The total surface area of a cube-shaped box is 216 cm2. What is its volume?

A cube with length 9 cm being cut into 27 smaller cubes. Find the total surface area of the small cubes and the volume for a small cube.

Xi Yuan has 1080 cm3 of syrup and five cubic containers of the same size. All the syrup is poured into the cubic containers until they are full. Find the length of each side of the cubic containers. The volume of each cubic container = …..... x ……… = ……………. cm3 The length of each side of the cubic container = ....................... = ………………………


17

The volume of cube is 64 m3. Find the surface are of the cube. Answer: 96 cm2

A rectangular block is 9 cm long, 4 cm wide and 6 cm high. Find the number of cubes which can be obtained from the block if the length of a cube is 3 cm. Answer: 8 cubes

A cubic water tank, 8 m in length, is half filled with water. Find the volume of the water. Answer: 256 m3

Computation involving addition, subtraction, multiplication, division and mixed operations on squares, square roots,

cubes and cube roots.

6 643

+ 36

= 6 4 + 6 = ……….+ ……… = ………..

4 − −643

= 4 − (−4)

= ……….. ………. = ………….

4 25 ÷ 22 = 4 5 ÷ 4 = ……… ÷ ……..

= …………..

5 2163

+ 121

5 − −1253

5 100 ÷ 52

5123

+42

9

1 −19

27

3

8

27

3×

1

4+

1

3

2


18

Given that 3.5482 = 12.59 and 1.1222 = 1.259, find the value of the following without using a calculator. a. 35.482

35.482 = ………….. x …………. = ………….. x ………….. = ……………

b. 1.259

1.259 = ………………

c. 12590

12590 = ……………………… . .

= ……………….. x ……………. = …………………. d. 0.11222 0.11222 = …………… ÷ ………….. = …………… ÷ ………….. = ……………

Given that 0.12 = 0.01, find the value of 102.

Given that 3000 = 54.77, find the

value of 30 + 0.3


value of 9604.

Find the value of 729.3

7293

= 273

= ………………

3𝑥 × 12𝑥 = 3600

3𝑥 × 12𝑥 = 3600

36𝑥2 = 3600

6𝑥 = 3600

𝑥 =3600

6

= 600

Given that 𝑥3

= 4, 𝑦3 = 3, calculate

the value x – y


27𝑥 × 3𝑥 = 3600

1233

× 1233

× 1233


19

ENRICHMENT EXERCISE:

27𝑥 × 3𝑥 = 3600

𝑥 + 3 − 2 = 7


Given a right-angled triangle with the height 5 and the

base 125

5. Find the area for the triangle.

2 + 2 2 − 2

Given that 50 = 7.071 and 5 = 2.236 , find the value

of 0.05 + 0.005.

Simplify 1

123

3

123

3×

2

123

3×

123

3×

4

123

3

Given that 5 = 2.237, find the value of

2.2372 + 1

2 × −8

3 ×

1

2

Answer: 4


20

Given that 𝑥 = 2, 𝑦 = 3, calculate the value 𝑥3 − 𝑦3

13 + 23 + 33 + ⋯+ 303 = x, find the value of x. Answer: 216225

2.4 ÷ 273

2

+ 2

3× 0.36

3

− 1

643 × 0.000008

3

Evaluate 2 − 172 − 1523

× 2× 18

2163

If x3 = 0.000064, find the value of 2 𝑥 + 0.963

7

125 × 3 × 12 ×

5

14÷ 2

10584 − 2 24 − 11 54 2

Answer: 22500

Answer: 96 cm

The diagram shows a rectangle PQRS and a square STUV, RST is a straight line. Given that PQ = UV and the area of STUV is 256 cm2, find the perimeter in cm, of the whole diagram.


21

Chapter 2 2tb2 Print

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