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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
NAME: ………………………………………………………. CLASS: 2TB2 DATE: ………………………
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
………………………………………………………………………………………………………………………
NAME: ………………………………………………………. CLASS: 2TB2 DATE: ………………………
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
NAME: ………………………………………………………. CLASS: 2TB2 DATE: ………………………
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
NAME: ………………………………………………………. CLASS: 2TB2 DATE: ………………………
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 =
NAME: ………………………………………………………. CLASS: 2TB2 DATE: ………………………
6
Estimating square roots of positive numbers
≈ means “ is approximately equal to”
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
NAME: ………………………………………………………. CLASS: 2TB2 DATE: ………………………
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 3.252 = 10.56, find the value of 32.52.
Given that 1.5122 = 2.286, find the value of 151.22.
Given that 3.912 = 15.29, find the value of 0.3912. 0.3912 = 3.912 ÷ 102
= ……………….. ÷………………
= ……………….
Given that 5.882 = 34.57, find the value of 0.05882. 0.05882 = 5.882 ÷ 1002
= …………..…… ÷ …………..…
= ……………….
Given that 1.582 = 2.496, find the value of 0.1582. 0.1582 = 1.582 ÷ 102
= …………..…… ÷ …………..…
= ……………….
Given that 3.612 = 13.03, find the value of 0.3612.
Given that 3.882 = 15.05, find the value of 0.03882.
Given that 6.582 = 43.29, find the value of 0.6582.
Given that 5.92 = 34.81, find the value of 5902.
Given that 0.2372 = 0.0562, find the value of 23.72.
Given that 15.82 = 249.646, find the value of 0.1582.
NAME: ………………………………………………………. CLASS: 2TB2 DATE: ………………………
8
Given that 2.5 = 1.581, then find
the value of 250.
250 = 2.5 × 100
= 1.581 x 10
= …………………
Given that 3.6 = 1.897, then find
the value of 360.
Given that 4.9 = 2.213, then find
the value of 490.
Given that 8.1 = 2.846, then find
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.
Given that 8.1 = 2.846, then find
the value of 0.081.
0.081 = 8.1 ÷ 100
= 2.846 ÷ 10
= ……………………..
Given that 10 = 3.162, then find
the value of 0.1.
0.1 = 10 ÷ 100
= …………. ÷ ……………..
= ……………………..
Given that 2 = 1.414, then find the
value of 0.02.
0.02 = 2 ÷ 100
= …………. ÷ ……………..
= ……………………..
Given that 2.5 = 1.581, then find
the value of 0.025.
Given that 3.6 = 1.897, then find
the value of 0.00036.
Given that 4.9 = 2.213, then find
the value of 0.00049.
Given that 2.5 = 1.581, then find
the value of 0.00025.
Given that 10 = 3.162, then find
the value of 0.001.
Given that 2 = 1.414, then find the
value of 20000.
NAME: ………………………………………………………. CLASS: 2TB2 DATE: ………………………
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.
NAME: ………………………………………………………. CLASS: 2TB2 DATE: ………………………
10
Given that 0.9502 = 0.9025, find the value of 95.02.
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 0.9302 = 0.8649, find the value of 93.02.
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
Given that 8.1 = 2.846, find the
value of 810 + 0.081
NAME: ………………………………………………………. CLASS: 2TB2 DATE: ………………………
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 =
NAME: ………………………………………………………. CLASS: 2TB2 DATE: ………………………
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
NAME: ………………………………………………………. CLASS: 2TB2 DATE: ………………………
13
41. −1.1 3 42. −0.7 3 43. −0.08 3 44. −0.07 3
Estimating cubes of numbers
≈ means “ is approximately equal to”
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.
Given that 53 = 125, find 1253
1253
= 533
= …………..
Given that 63 = 216, find 2163
Given that 73 = 343, find 3433
NAME: ………………………………………………………. CLASS: 2TB2 DATE: ………………………
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
NAME: ………………………………………………………. CLASS: 2TB2 DATE: ………………………
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
≈ means “ is approximately equal to”
−73
≈ −83
−73
≈ …………….
−263
713
2123
126.93
7.43
283
−1393
0.2313
NAME: ………………………………………………………. CLASS: 2TB2 DATE: ………………………
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 = ....................... = ………………………
NAME: ………………………………………………………. CLASS: 2TB2 DATE: ………………………
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
NAME: ………………………………………………………. CLASS: 2TB2 DATE: ………………………
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
Given that 0.9802 = 0.9604, find the
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
Find the value of 64.3
27𝑥 × 3𝑥 = 3600
1233
× 1233
× 1233
NAME: ………………………………………………………. CLASS: 2TB2 DATE: ………………………
19
ENRICHMENT EXERCISE:
27𝑥 × 3𝑥 = 3600
𝑥 + 3 − 2 = 7
Find the value of 4096.3
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
NAME: ………………………………………………………. CLASS: 2TB2 DATE: ………………………
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.
NAME: ………………………………………………………. CLASS: 2TB2 DATE: ………………………
21