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What is a Perfect Cube?
Grade Level:
Class 7
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
A perfect cube is a number that can be obtained by multiplying an integer by itself three times. In simpler terms, if you multiply a whole number (like 1, 2, 3, etc.) by itself, and then multiply the result by the same number again, you get a perfect cube.
Simple Example
Quick Example
Imagine you have a Rubik's Cube. Each side has 3 small squares. If you want to know the total number of tiny cubes that make up the whole Rubik's Cube, you'd multiply 3 x 3 x 3. This gives you 27. So, 27 is a perfect cube because it's the result of 3 multiplied by itself three times.
Worked Example
Step-by-Step
Let's find out if 64 is a perfect cube.
1. First, we try to find a number that, when multiplied by itself three times, gives 64.
---2. We can start by testing small whole numbers.
---3. Try 1: 1 x 1 x 1 = 1 (Not 64)
---4. Try 2: 2 x 2 x 2 = 8 (Not 64)
---5. Try 3: 3 x 3 x 3 = 27 (Not 64)
---6. Try 4: 4 x 4 x 4 = 64 (Yes, it is!)
---So, 64 is a perfect cube because it is the result of 4 multiplied by itself three times. The cube root of 64 is 4.
Why It Matters
Understanding perfect cubes helps in fields like computer science for memory allocation and in physics for calculating volumes. Engineers use these concepts when designing structures, and even in data science, understanding powers helps in analyzing patterns. It's a foundational skill for many exciting careers!
Common Mistakes
MISTAKE: Confusing a perfect cube with a perfect square (multiplying a number by itself twice). For example, thinking 9 is a perfect cube because 3 x 3 = 9. | CORRECTION: Remember a perfect cube means multiplying the number by itself THREE times. 9 is a perfect square, not a perfect cube. (3 x 3 x 3 = 27, so 27 is a perfect cube).
MISTAKE: Only checking if a number is divisible by 3. For example, thinking 12 is a perfect cube because it's divisible by 3. | CORRECTION: Being divisible by 3 does not make a number a perfect cube. You need to find a whole number that, when multiplied by itself three times, gives the original number.
MISTAKE: Incorrectly calculating the product. For example, calculating 5 x 5 x 5 as 75. | CORRECTION: Always do the multiplication carefully: 5 x 5 = 25, then 25 x 5 = 125. Double-check your arithmetic.
Practice Questions
Try It Yourself
QUESTION: Is 8 a perfect cube? If yes, what number is multiplied by itself three times to get 8? | ANSWER: Yes, 8 is a perfect cube. 2 x 2 x 2 = 8.
QUESTION: Find the perfect cube between 100 and 150. | ANSWER: 125 (because 5 x 5 x 5 = 125)
QUESTION: A small tiffin box has a side length of 6 cm. If you wanted to arrange smaller cube-shaped chocolates, each with a side length of 2 cm, how many chocolates would fit inside the tiffin box? (Hint: Find the volume of the tiffin box and one chocolate first). | ANSWER: Volume of tiffin box = 6 x 6 x 6 = 216 cubic cm. Volume of one chocolate = 2 x 2 x 2 = 8 cubic cm. Number of chocolates = 216 / 8 = 27 chocolates.
MCQ
Quick Quiz
Which of the following numbers is NOT a perfect cube?
1
27
64
50
The Correct Answer Is:
D
1 is 1x1x1, 27 is 3x3x3, and 64 is 4x4x4. 50 cannot be obtained by multiplying a whole number by itself three times, making it not a perfect cube.
Real World Connection
In the Real World
In India, think about how water tanks are designed. If a tank is shaped like a cube, knowing its side length helps engineers calculate its total water holding capacity (volume). This is a real-world application of perfect cubes, ensuring our homes have enough water supply!
Key Vocabulary
Key Terms
PERFECT CUBE: A number obtained by multiplying an integer by itself three times | CUBE ROOT: The number that, when multiplied by itself three times, gives the perfect cube | INTEGER: A whole number (positive, negative, or zero) | VOLUME: The amount of space a 3D object occupies
What's Next
What to Learn Next
Great job understanding perfect cubes! Next, you can explore 'Cube Roots'. This will teach you how to find the original number when you are given a perfect cube, which is super useful for solving many math problems.
