Squares, Square Roots, Cubes and Cube Roots: Mathematics Tingkatan 1 questions

Section A: objective questions

12 marks

Choose the correct answer.

Which of the following is a perfect square?

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49 = 7 × 7 = 7². 18, 27 and 50 are not the square of any whole number. (27 is a perfect cube, not a perfect square.)

For the teacher: Remembering, low difficulty, standard 3.1.1

Textbook p. 49

Diagram 1 shows the first four patterns of a sequence. Each pattern is a square of dots. How many dots are there in Pattern 6?

Pattern 1: 1 dot; Pattern 2: 2 by 2 dots; Pattern 3: 3 by 3 dots; Pattern 4: 4 by 4 dots — Diagram 1

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Pattern n has n rows of n dots, so Pattern 6 has 6² = 6 × 6 = 36 dots. 12 is 6 × 2; 25 and 49 are Patterns 5 and 7.

For the teacher: Understanding, low difficulty, standard 3.1.1

Textbook p. 48

(−0.4)² = ______

−0.16
0.16
0.8
0.016

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(−0.4) × (−0.4) = 0.16. Negative × negative is positive, and 1 + 1 = 2 decimal places. 0.8 is 0.4 × 2.

For the teacher: Applying, low difficulty, standard 3.1.4

Textbook p. 52

√0.49 = ______

0.07
7
0.245
0.7

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0.7 × 0.7 = 0.49, so √0.49 = 0.7. 0.07 × 0.07 = 0.0049, and 0.245 is 0.49 ÷ 2.

For the teacher: Applying, medium difficulty, standard 3.1.5

Textbook p. 54

Which point on the number line in Diagram 2 best shows the position of √70?

Number line from 7 to 10 marked in tenths. P is at 7.6, Q at 8.4, R at 8.9 and S at 9.6 — Diagram 2

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70 is between the perfect squares 64 and 81, so √70 is between 8 and 9. 70 is much nearer 64 than 81, so √70 is a little more than 8 (it is 8.37). Q at 8.4 is the best.

For the teacher: Understanding, medium difficulty, standard 3.1.7

Textbook p. 55

Which of the following has the same value as 4³?

4 × 3
4 + 4 + 4
4 × 4 × 4
3 × 3 × 3 × 3

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4³ means 4 multiplied by itself three times: 4 × 4 × 4 = 64. 4 × 3 and 4 + 4 + 4 are both 12.

For the teacher: Remembering, low difficulty, standard 3.2.1

Textbook p. 58

(−3)³ = ______

−27
27
−9
9

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(−3) × (−3) × (−3) = 9 × (−3) = −27. The cube of a negative number is negative. −9 is (−3) × 3.

For the teacher: Applying, low difficulty, standard 3.2.4

Textbook p. 63

³√−0.064 = ______

0.4
−0.04
0.04
−0.4

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(−0.4) × (−0.4) × (−0.4) = −0.064, so the answer is −0.4. The cube root of a negative number is negative. (−0.04)³ = −0.000064.

For the teacher: Applying, medium difficulty, standard 3.2.5

Textbook p. 65

Diagram 3 shows the factor tree of 729. Which statement about 729 is correct?

Factor tree: 729 = 9 × 81, 9 = 3 × 3, 81 = 9 × 9, each 9 = 3 × 3 — Diagram 3

It is a perfect square only
It is a perfect square and a perfect cube
It is a perfect cube only
It is neither a perfect square nor a perfect cube

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729 = 3 × 3 × 3 × 3 × 3 × 3. Six 3s can be split into two identical groups, (3 × 3 × 3)² = 27², and into three identical groups, (3 × 3)³ = 9³. So 729 is both.

For the teacher: Analysing, medium difficulty, standard 3.2.2

Textbook p. 60

Estimate the value of ³√70.

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70 is between the perfect cubes 64 = 4³ and 125 = 5³. 70 is much nearer 64, so ³√70 ≈ 4. 23 comes from 70 ÷ 3, which is wrong.

For the teacher: Applying, medium difficulty, standard 3.2.7

Textbook p. 66

³√−8 × 3² − √16 = ______

−14
22
−10
−22

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−2 × 9 − 4 = −18 − 4 = −22. −14 comes from −18 + 4; −10 from doing the subtraction first, −2 × (9 − 4).

For the teacher: Applying, high difficulty, standard 3.2.9

Textbook p. 68

√3 × √12 = ______

√15
36
6
15

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√a × √b = √ab: √3 × 12 = √36 = 6. √15 adds the numbers instead of multiplying; 36 forgets the square root.

For the teacher: Understanding, medium difficulty, standard 3.1.8

Textbook p. 55

Section B: structured questions

28 marks

Answer all questions. Show your working. Use a calculator only in Question 3.

Diagram 4 shows the factor tree of 324.

Factor tree: 324 = 4 × 81, 4 = 2 × 2, 81 = 9 × 9, each 9 = 3 × 3 — Diagram 4

[1 mark] (a)Write 324 as a product of its prime factors.

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324 = 2 × 2 × 3 × 3 × 3 × 3 [1]1

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[1 mark] (b)Is 324 a perfect square? Give a reason.

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Yes. The prime factors can be split into two identical groups: (2 × 3 × 3) × (2 × 3 × 3) [1]1

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[1 mark] (c)Hence, find √324.

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√324 = 2 × 3 × 3 = 18 [1]1

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For the teacher: Understanding, low difficulty, standard 3.1.2

Textbook p. 50

Find the value of each of the following without using a calculator.

[2 marks] (a)(−113)²

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(−43) × (−43) [1]1
= 169 = 179 [1]1

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[2 marks] (b)³√338

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³√278 [1]1
= 32 = 112 [1]1

Change the mixed number to an improper fraction first. 27 = 3³ and 8 = 2³.

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For the teacher: Applying, medium difficulty, standard 3.1.5

Textbook p. 53

Use a calculator to find the value of each of the following.

[1 mark] (a)√57.6, correct to two decimal places

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√57.6 = 7.589… = 7.59 [1]1

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[1 mark] (b)³√−45, correct to two decimal places

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³√−45 = −3.556… = −3.56 [1]1

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[1 mark] (c)2.35³

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2.35³ = 12.977875 [1]1

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For the teacher: Applying, low difficulty, standard 3.1.6

Textbook p. 54

Diagram 5 shows a square photo with an area of 110.25 cm² pasted in the middle of a square cardboard with sides of 15 cm.

Square cardboard with sides of 15 cm; a square photo of area 110.25 cm² in the middle — Diagram 5

[1 mark] (a)Find the length of the side of the photo.

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√110.25 = 10.5 cm [1]1

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[2 marks] (b)How far is the photo from each side of the cardboard?

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15 − 10.5 = 4.5 [1]1
4.5 ÷ 2 = 2.25 cm [1]1

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[1 mark] (c)Find the area of the cardboard that is not covered by the photo.

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15² − 110.25 = 225 − 110.25 = 114.75 cm² [1]1

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For the teacher: Analysing, medium difficulty, standard 3.1.9

Textbook p. 56

Diagram 6 shows a wooden cube with sides of 8 cm. A small cube is cut out of one corner. The volume of the remaining block is 485 cm³.

Wooden cube of side 8 cm with a small cube cut from the top corner; remaining block 485 cm³ — Diagram 6

[1 mark] (a)Find the volume of the wooden cube before the small cube was cut out.

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8³ = 512 cm³ [1]1

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[2 marks] (b)Find the length of the side of the small cube.

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512 − 485 = 27 cm³ [1]1
³√27 = 3 cm [1]1

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For the teacher: Applying, medium difficulty, standard 3.2.8

Textbook p. 67

Ravi has a piece of wire 150 cm long. He wants to bend it into the frame of a cube, as in Diagram 7. The volume of the cube must be 1 728 cm³.

Wire frame of a cube; volume of the cube is 1 728 cm³ — Diagram 7

[1 mark] (a)Find the length of each edge of the cube.

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³√1 728 = 12 cm [1]1

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[2 marks] (b)Is Ravi's wire long enough? Give a reason with your working.

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A cube has 12 edges: 12 × 12 = 144 cm [1]1
Yes, because 144 cm is less than 150 cm [1]1

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[1 mark] (c)How much wire is left over?

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150 − 144 = 6 cm [1]1

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For the teacher: Analysing, high difficulty, standard 3.2.8

Textbook p. 67

Hafiz worked out 5² − ³√−64 × √9. Diagram 8 shows his working.

Hafiz's working: 5² − cube root of −64 × square root of 9; Line 1: = 25 − (−4) × 3; Line 2: = 29 × 3; Line 3: = 87 — Diagram 8

[2 marks] (a)Hafiz's answer is wrong. In which line did he make a mistake? Explain the mistake.

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Line 2 [1]1
He subtracted before multiplying. × must be done before − [1]1

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[2 marks] (b)Find the correct value.

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25 − (−12) or 25 + 12 [1]1
= 37 [1]1

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For the teacher: Evaluating, high difficulty, standard 3.2.9

Textbook p. 68

100 members of a cultural club march in the Citrawarna parade. They stand in square formations, with the same number of members in every row and every column.

[1 mark] (a)All 100 members form one square. How many members are there in each row?

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√100 = 10 members [1]1

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[2 marks] (b)Later, the members split up to form two squares at the same time, using all 100 members. Design the two squares: state the number of members in each row of each square and show that your design works.

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Two perfect squares that add up to 100: 36 + 64 = 100 [1]1
One square with 6 in each row and one square with 8 in each row [1]1

Try the perfect squares 1, 4, 9, 16, 25, 36, 49, 64, 81 and look for a pair that adds up to 100. Only 36 and 64 work.

Tick each point you got.

For the teacher: Creating, high difficulty, standard 3.1.9

Textbook p. 57

Section A: objective questions

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Section B: structured questions

Question 13

Question 14

Question 15

Question 16

Question 17

Question 18

Question 19

Question 20