Worksheet - Calculations with Square Roots and Cube Roots of integers

Table of Contents
1. Instructions to the Learner
2. Section A: Multiple Choice
3. Section B: Short Answer and Structured Questions
4. Section C: Problem Solving and Word Problems
5. Bonus / Challenge Question
6. Answer Key
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Instructions to the Learner

1. Read all questions carefully before answering.
2. Show all your working clearly. Marks are awarded for correct methods and calculations.
3. Write your final answers clearly and circle or underline them where appropriate.
4. Use a calculator only where specifically instructed to do so.
5. All answers involving square roots and cube roots should be simplified where possible.
6. Attempt all questions. If you finish early, check your work and attempt the bonus question.

Section A: Multiple Choice

Q1. What is the value of \(\sqrt{64}\)? [2 marks]

1. 6
2. 7
3. 8
4. 32

Q2. Calculate \(\sqrt[3]{27}\). [2 marks]

1. 3
2. 9
3. 13.5
4. 27

Q3. What is \(\sqrt{100} + \sqrt{25}\)? [2 marks]

1. 11
2. 12
3. 15
4. 125

Q4. Which of the following is equal to \(\sqrt[3]{-8}\)? [2 marks]

1. -2
2. 2
3. -4
4. Not defined

Q5. Calculate \(\sqrt{144} - \sqrt[3]{64}\). [2 marks]

1. 4
2. 6
3. 8
4. 10

Q6. What is the value of \(\sqrt{49} × \sqrt{4}\)? [2 marks]

1. 14
2. 28
3. 53
4. 196

Section B: Short Answer and Structured Questions

Q1. Calculate the following: [6 marks]

• (a) \(\sqrt{81}\) [2 marks]
• (b) \(\sqrt[3]{125}\) [2 marks]
• (c) \(\sqrt{16} + \sqrt[3]{8}\) [2 marks]

Q2. Simplify the following expressions: [6 marks]

• (a) \(\sqrt{36} - \sqrt{9}\) [2 marks]
• (b) \(\sqrt[3]{1} + \sqrt{1}\) [2 marks]
• (c) \(2 × \sqrt{25}\) [2 marks]

Q3. Evaluate: [8 marks]

• (a) \(\sqrt{169}\) [2 marks]
• (b) \(\sqrt[3]{-27}\) [2 marks]
• (c) \(\sqrt{121} ÷ \sqrt{11}\) [2 marks]
• (d) \(\sqrt[3]{1000} - \sqrt{100}\) [2 marks]

Q4. Calculate the following, showing all steps: [9 marks]

• (a) \(\sqrt{225} + \sqrt{16}\) [3 marks]
• (b) \(\sqrt[3]{216} × \sqrt{4}\) [3 marks]
• (c) \((\sqrt{36})^2 - \sqrt[3]{64}\) [3 marks]

Section C: Problem Solving and Word Problems

Q1. Thabo is designing a square garden for his school project. [7 marks]

• (a) If the area of the garden is 196 m², calculate the length of one side using square roots. [3 marks]
• (b) Thabo wants to build a fence around the garden. Calculate the total perimeter. [2 marks]
• (c) If he decides to double the area, what will the new side length be? [2 marks]

Q2. Nomvula has a cube-shaped storage box. [8 marks]

• (a) If the volume of the box is 512 cm³, use cube roots to find the length of one edge of the box. [3 marks]
• (b) Calculate the total surface area of one face of the cube. [2 marks]
• (c) If Nomvula stacks three identical boxes on top of each other, what is the total height of the stack? [3 marks]

Q3. Sipho and Lerato are comparing their mathematics calculations. [7 marks]

• (a) Sipho calculates \(\sqrt{400} + \sqrt[3]{125}\). Show his working and find his answer. [3 marks]
• (b) Lerato calculates \(\sqrt{289} - \sqrt{49}\). Show her working and find her answer. [2 marks]
• (c) Who obtained the larger answer and by how much? [2 marks]

Bonus / Challenge Question

Optional question for fast finishers:

Q1. Consider the following calculations involving square roots and cube roots: [6 marks]

• (a) Calculate \(\sqrt{16} + \sqrt{36} + \sqrt{64} + \sqrt{100}\). [3 marks]
• (b) Calculate \(\sqrt[3]{8} + \sqrt[3]{27} + \sqrt[3]{64}\). [2 marks]
• (c) Which sum is larger and by how much? [1 mark]

Answer Key

Section A: Multiple Choice

Q1. C - 8

Explanation: \(\sqrt{64} = 8\) because 8 × 8 = 64. Option A (6) is incorrect because 6 × 6 = 36. Option B (7) is incorrect because 7 × 7 = 49. Option D (32) is incorrect as it represents 64 ÷ 2, not the square root.

Q2. A - 3

Explanation: \(\sqrt[3]{27} = 3\) because 3 × 3 × 3 = 27. Option B (9) is incorrect because 9 × 9 × 9 = 729. Option C (13.5) is incorrect as it represents 27 ÷ 2. Option D (27) is the original number, not its cube root.

Q3. C - 15

Explanation: \(\sqrt{100} = 10\) and \(\sqrt{25} = 5\), so 10 + 5 = 15. Option A (11) is incorrect. Option B (12) is incorrect. Option D (125) incorrectly suggests multiplication of the original values.

Q4. A - -2

Explanation: \(\sqrt[3]{-8} = -2\) because (-2) × (-2) × (-2) = -8. Cube roots of negative numbers are defined and negative. Option B (2) is incorrect as it gives a positive value. Option C (-4) is incorrect. Option D is incorrect because cube roots of negative numbers exist.

Q5. C - 8

Explanation: \(\sqrt{144} = 12\) and \(\sqrt[3]{64} = 4\), so 12 - 4 = 8. Option A (4) represents only the cube root value. Option B (6) is incorrect. Option D (10) is incorrect.

Q6. A - 14

Explanation: \(\sqrt{49} = 7\) and \(\sqrt{4} = 2\), so 7 × 2 = 14. Option B (28) represents 7 × 4 or 14 × 2. Option C (53) is incorrect. Option D (196) represents 49 × 4, not the product of the square roots.

Section B: Short Answer and Structured Questions

Q1.

(a) \(\sqrt{81}\) [2 marks]
\(\sqrt{81} = 9\) (1 mark for method, 1 mark for answer)
Because 9 × 9 = 81
Answer: 9

(b) \(\sqrt[3]{125}\) [2 marks]
\(\sqrt[3]{125} = 5\) (1 mark for method, 1 mark for answer)
Because 5 × 5 × 5 = 125
Answer: 5

(c) \(\sqrt{16} + \sqrt[3]{8}\) [2 marks]
\(\sqrt{16} = 4\) (½ mark)
\(\sqrt[3]{8} = 2\) (½ mark)
4 + 2 = 6 (1 mark)
Answer: 6

Q2.

(a) \(\sqrt{36} - \sqrt{9}\) [2 marks]
\(\sqrt{36} = 6\) (½ mark)
\(\sqrt{9} = 3\) (½ mark)
6 - 3 = 3 (1 mark)
Answer: 3

(b) \(\sqrt[3]{1} + \sqrt{1}\) [2 marks]
\(\sqrt[3]{1} = 1\) (½ mark)
\(\sqrt{1} = 1\) (½ mark)
1 + 1 = 2 (1 mark)
Answer: 2

(c) \(2 × \sqrt{25}\) [2 marks]
\(\sqrt{25} = 5\) (1 mark)
2 × 5 = 10 (1 mark)
Answer: 10

Q3.

(a) \(\sqrt{169}\) [2 marks]
\(\sqrt{169} = 13\) (1 mark for method, 1 mark for answer)
Because 13 × 13 = 169
Answer: 13

(b) \(\sqrt[3]{-27}\) [2 marks]
\(\sqrt[3]{-27} = -3\) (1 mark for method, 1 mark for answer)
Because (-3) × (-3) × (-3) = -27
Answer: -3

(c) \(\sqrt{121} ÷ \sqrt{11}\) [2 marks]
\(\sqrt{121} = 11\) (½ mark)
\(\sqrt{11} = \sqrt{11}\) (½ mark)
\(11 ÷ \sqrt{11} = \sqrt{11}\) (1 mark)
Answer: \(\sqrt{11}\) or approximately 3.32

(d) \(\sqrt[3]{1000} - \sqrt{100}\) [2 marks]
\(\sqrt[3]{1000} = 10\) (½ mark)
\(\sqrt{100} = 10\) (½ mark)
10 - 10 = 0 (1 mark)
Answer: 0

Q4.

(a) \(\sqrt{225} + \sqrt{16}\) [3 marks]
\(\sqrt{225} = 15\) (1 mark)
\(\sqrt{16} = 4\) (1 mark)
15 + 4 = 19 (1 mark)
Answer: 19

(b) \(\sqrt[3]{216} × \sqrt{4}\) [3 marks]
\(\sqrt[3]{216} = 6\) because 6 × 6 × 6 = 216 (1 mark)
\(\sqrt{4} = 2\) (1 mark)
6 × 2 = 12 (1 mark)
Answer: 12

(c) \((\sqrt{36})^2 - \sqrt[3]{64}\) [3 marks]
\(\sqrt{36} = 6\) (½ mark)
\((6)^2 = 36\) (1 mark)
\(\sqrt[3]{64} = 4\) (½ mark)
36 - 4 = 32 (1 mark)
Answer: 32

Section C: Problem Solving and Word Problems

Q1. Thabo's garden problem

(a) Finding the side length [3 marks]
Area of square = side × side = side²
side² = 196 m² (1 mark)
side = \(\sqrt{196}\) (1 mark)
side = 14 m (1 mark)
Answer: The side length is 14 m

(b) Calculating the perimeter [2 marks]
Perimeter = 4 × side (½ mark)
Perimeter = 4 × 14 (½ mark)
Perimeter = 56 m (1 mark)
Answer: Thabo needs 56 m of fencing

(c) New side length with doubled area [2 marks]
New area = 2 × 196 = 392 m² (½ mark)
New side = \(\sqrt{392}\) (½ mark)
New side = 14\(\sqrt{2}\) m ≈ 19.8 m (1 mark)
Answer: The new side length is 14√2 m or approximately 19.8 m

Q2. Nomvula's storage box problem

(a) Finding the edge length [3 marks]
Volume of cube = edge × edge × edge = edge³
edge³ = 512 cm³ (1 mark)
edge = \(\sqrt[3]{512}\) (1 mark)
edge = 8 cm (1 mark)
Answer: The edge length is 8 cm

(b) Surface area of one face [2 marks]
Area of one face = edge × edge (½ mark)
Area = 8 × 8 (½ mark)
Area = 64 cm² (1 mark)
Answer: The surface area of one face is 64 cm²

(c) Total height of three boxes [3 marks]
One box height = 8 cm (1 mark)
Three boxes height = 3 × 8 (1 mark)
Total height = 24 cm (1 mark)
Answer: The total height of Nomvula's stack is 24 cm

Q3. Sipho and Lerato's calculations

(a) Sipho's calculation [3 marks]
\(\sqrt{400} = 20\) (1 mark)
\(\sqrt[3]{125} = 5\) (1 mark)
20 + 5 = 25 (1 mark)
Answer: Sipho's answer is 25

(b) Lerato's calculation [2 marks]
\(\sqrt{289} = 17\) (½ mark)
\(\sqrt{49} = 7\) (½ mark)
17 - 7 = 10 (1 mark)
Answer: Lerato's answer is 10

(c) Comparing their answers [2 marks]
Sipho's answer = 25 (given)
Lerato's answer = 10 (given)
25 - 10 = 15 (1 mark)
Sipho obtained the larger answer (1 mark)
Answer: Sipho obtained the larger answer by 15

Bonus / Challenge Question

Q1.

(a) \(\sqrt{16} + \sqrt{36} + \sqrt{64} + \sqrt{100}\) [3 marks]
\(\sqrt{16} = 4\) (½ mark)
\(\sqrt{36} = 6\) (½ mark)
\(\sqrt{64} = 8\) (½ mark)
\(\sqrt{100} = 10\) (½ mark)
4 + 6 + 8 + 10 = 28 (1 mark)
Answer: 28

(b) \(\sqrt[3]{8} + \sqrt[3]{27} + \sqrt[3]{64}\) [2 marks]
\(\sqrt[3]{8} = 2\) (½ mark)
\(\sqrt[3]{27} = 3\) (½ mark)
\(\sqrt[3]{64} = 4\) (½ mark)
2 + 3 + 4 = 9 (½ mark)
Answer: 9

(c) Comparing the sums [1 mark]
28 - 9 = 19
Answer: The sum of square roots (28) is larger by 19

Note: This question tests the ability to work with multiple square roots and cube roots sequentially and compare cumulative results, requiring sustained concentration and systematic calculation skills.

Total Marks Summary