Perfect Cubes - Interactive Learning
Welcome to this interactive lesson on perfect cubes! Explore the concepts through questions, visualize 3D shapes, and discover solutions.
Question 1: Which numbers are not perfect cubes?
Given:
Numbers: 216, 128, 1000, 100
To Find:
Which of these numbers are not perfect cubes?
Solution:
A perfect cube is a number that can be expressed as the cube of an integer.
• 216 = 6 × 6 × 6 = 6³ → Perfect cube
• 128 = 2 × 2 × 2 × 2 × 2 × 2 × 2 = (2³) × (2³) × 2 = 8 × 8 × 2 → Not a perfect cube
• 1000 = 10 × 10 × 10 = 10³ → Perfect cube
• 100 = 10 × 10 = 10² → Not a perfect cube
Answer: 128 and 100 are not perfect cubes.
Question 2: Smallest multiplier for perfect cube
Given:
Numbers: 243, 256, 72, 675, 100
To Find:
The smallest number to multiply each by to obtain a perfect cube
Solution:
We need to find the prime factors and make each exponent a multiple of 3.
• 243 = 3 × 3 × 3 × 3 × 3 = 3⁵ → Need 3¹ to make 3⁶ → Multiply by 3
• 256 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2⁸ → Need 2¹ to make 2⁹ → Multiply by 2
• 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3² → Need 3¹ to make 3³ → Multiply by 3
• 675 = 3 × 3 × 3 × 5 × 5 = 3³ × 5² → Need 5¹ to make 5³ → Multiply by 5
• 100 = 2 × 2 × 5 × 5 = 2² × 5² → Need 2¹ × 5¹ to make 2³ × 5³ → Multiply by 10
Question 3: Smallest divisor for perfect cube
Given:
Numbers: 81, 128, 135, 192, 704
To Find:
The smallest number to divide each by to obtain a perfect cube
Solution:
We need to find the prime factors and remove extra factors so that each exponent is a multiple of 3.
• 81 = 3 × 3 × 3 × 3 = 3⁴ → Remove 3¹ to get 3³ → Divide by 3
• 128 = 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2⁷ → Remove 2¹ to get 2⁶ → Divide by 2
• 135 = 3 × 3 × 3 × 5 = 3³ × 5¹ → Remove 5¹ to get 3³ → Divide by 5
• 192 = 2 × 2 × 2 × 2 × 2 × 2 × 3 = 2⁶ × 3¹ → Remove 3¹ to get 2⁶ → Divide by 3
• 704 = 2 × 2 × 2 × 2 × 2 × 2 × 11 = 2⁶ × 11¹ → Remove 11¹ to get 2⁶ → Divide by 11
Question 4: Cuboids to form a cube
Given:
Parikshit makes a cuboid of plasticine with sides 5 cm, 2 cm, 5 cm.
To Find:
How many such cuboids are needed to form a cube?
Solution:
Volume of one cuboid = 5 × 2 × 5 = 50 cm³
To form a cube, the volume must be a perfect cube.
We need to find the smallest perfect cube that is a multiple of 50.
50 = 2 × 5 × 5 = 2¹ × 5²
To make a perfect cube, we need 2² × 5¹ = 4 × 5 = 20
So we need 20 cuboids to form a cube.
Volume of cube = 20 × 50 = 1000 cm³ = 10³ cm³
Side of cube = ∛1000 = 10 cm
3D Visualization
Perfect Cube (All sides equal)
Cuboid (Question 4: 5×2×5)
Previous Knowledge Required
- Understanding of prime factorization
- Knowledge of exponents and powers
- Concept of volume for 3D shapes
- Basic multiplication and division
