Worksheet for Chapter 7 "Cubes and Cube Roots" from Maths-VIII (E), including definitions and practice questions.

I'll solve each question step by step.

Question 1: Find the cube of (-2/3)
- Cube means multiply the number by itself three times
- (-2/3)³ = (-2/3) × (-2/3) × (-2/3)
- Numerator: (-2) × (-2) × (-2) = -8
- Denominator: 3 × 3 × 3 = 27
- Answer: -8/27

Question 2: How much is (3/5)³ - (1/5)³?
- (3/5)³ = 3³/5³ = 27/125
- (1/5)³ = 1³/5³ = 1/125
- 27/125 - 1/125 = 26/125

Question 3: Simplify (0.5)³ + (0.2)³
- (0.5)³ = 0.5 × 0.5 × 0.5 = 0.125
- (0.2)³ = 0.2 × 0.2 × 0.2 = 0.008
- 0.125 + 0.008 = 0.133

Question 4: Find the volume of a cube of side 6 cm
- Volume of cube = side³
- Volume = 6³ = 6 × 6 × 6 = 216 cm³

Question 5: How many unit cubes can be formed by melting a copper cube of side 5 cm?
- Volume of large cube = 5³ = 125 cm³
- Each unit cube has volume 1 cm³
- Number of unit cubes = 125 ÷ 1 = 125

Question 6: Simplify 1 + (3/5)³
- (3/5)³ = 27/125
- 1 + 27/125 = 125/125 + 27/125 = 152/125

Question 7: If a number is written as 3 × 3 × 5 × 3 × 5 × 7 × 7, find the smallest number to multiply to form a perfect cube
- Rearranging: 3 × 3 × 3 × 5 × 5 × 7 × 7
- For a perfect cube, each prime factor must appear in groups of 3
- We have: 3³ ✓, 5² ✗, 7² ✗
- Need one more 5 and one more 7
- Smallest number = 5 × 7 = 35

Question 8: Find ∛(5 × 7 × 7 × 5 × 7 × 5)
- Rearranging: ∛(5 × 5 × 5 × 7 × 7 × 7) = ∛(5³ × 7³)
- = 5 × 7 = 35

Question 9: (0.3)³
- 0.3 × 0.3 × 0.3 = 0.027

Question 10: Find the value of ∛0.064
- 0.064 = 64/1000
- ∛64 = 4 (since 4³ = 64)
- ∛1000 = 10 (since 10³ = 1000)
- 0.064 = 4/10 = 0.4

Question 11: ∛(-216) - ∛125
- ∛(-216) = -6 (since (-6)³ = -216)
- ∛125 = 5 (since 5³ = 125)
- -6 - 5 = -11

Question 12: Cubes of positive numbers are always _______
- When you multiply a positive number by itself three times, the result is always positive
- Answer: positive

Question 13: Cube roots of positive numbers are always _______
- The cube root of a positive number is always positive
- Answer: positive

Final Answer:
1. -8/27
2. 26/125
3. 0.133
4. 216 cm³
5. 125
6. 152/125
7. 35
8. 35
9. 0.027
10. 0.4
11. -11
12. positive
13. positive