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What is the Properties of Cube Numbers?
Grade Level:
Class 7
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
Cube numbers are special numbers that you get by multiplying a whole number by itself three times. For example, 8 is a cube number because 2 x 2 x 2 = 8. Understanding their properties helps us work with them easily.
Simple Example
Quick Example
Imagine you have a Rubik's Cube. If each side has 3 small cubes, then the total number of small cubes in the big Rubik's Cube is 3 x 3 x 3 = 27. So, 27 is a cube number.
Worked Example
Step-by-Step
Let's find the properties of the cube number 64.
Step 1: First, identify the number. It is 64.
---Step 2: Find the prime factors of 64. 64 = 2 x 32 = 2 x 2 x 16 = 2 x 2 x 2 x 8 = 2 x 2 x 2 x 2 x 4 = 2 x 2 x 2 x 2 x 2 x 2.
---Step 3: Group the prime factors in threes (triplets). 64 = (2 x 2 x 2) x (2 x 2 x 2).
---Step 4: This shows that 64 is a perfect cube because all its prime factors can be grouped into triplets. Here, 64 = 4 x 4 x 4.
---Step 5: Observe its ending digit. 64 ends in 4. The cube root of 64 is 4, which also ends in 4. This is a property: a number ending in 4 will have its cube root ending in 4.
---Step 6: Also, 64 is an even number. Its cube root, 4, is also an even number. This is another property: the cube of an even number is always even.
Answer: 64 is a perfect cube (4^3), it's an even number, and its cube root ends in the same digit (4) as the number itself.
Why It Matters
Common Mistakes
MISTAKE: Confusing cube numbers with square numbers (e.g., thinking 27 is 9 squared). | CORRECTION: Remember that cube numbers are multiplied by themselves three times (n x n x n), while square numbers are multiplied by themselves two times (n x n).
MISTAKE: Assuming all numbers ending in 0, 1, 4, 5, 6, 9 are perfect cubes. | CORRECTION: While some perfect cubes end in these digits, it's not true for all numbers. For example, 10 is not a perfect cube, but it ends in 0. You must check by prime factorization.
MISTAKE: Incorrectly stating that the cube of an odd number is always even. | CORRECTION: The cube of an odd number is always odd (e.g., 3^3 = 27, 5^3 = 125). The cube of an even number is always even.
Practice Questions
Try It Yourself
QUESTION: Is 125 a perfect cube? If yes, find its cube root. | ANSWER: Yes, 125 is a perfect cube. Its cube root is 5 (5 x 5 x 5 = 125).
QUESTION: What is the last digit of the cube of 13? | ANSWER: The last digit of 13 is 3. The last digit of 3^3 (which is 27) is 7. So, the last digit of the cube of 13 is 7.
QUESTION: A number when cubed gives 216. Is this number odd or even? Explain why without calculating the cube root. | ANSWER: The number is even. This is because the cube of an even number is always even, and 216 is an even number.
MCQ
Quick Quiz
Which of the following statements about cube numbers is true?
The cube of an odd number is always even.
All numbers ending in 8 are perfect cubes.
The cube of an even number is always even.
A number multiplied by 3 gives its cube.
The Correct Answer Is:
C
The cube of an even number is always even (e.g., 2^3=8, 4^3=64). Option A is false as the cube of an odd number is odd. Option B is false; for example, 18 is not a perfect cube. Option D describes multiplication, not cubing.
Real World Connection
In the Real World
When you're designing a new building or a water tank in India, architects and civil engineers often need to calculate the volume. If the tank is cube-shaped, knowing cube numbers and their properties helps them quickly determine how much water it can hold or how much material is needed for construction. This is also important in video games for rendering 3D objects.
Key Vocabulary
Key Terms
CUBE NUMBER: A number obtained by multiplying an integer by itself three times (n x n x n) | PERFECT CUBE: A number that is the cube of an integer | PRIME FACTORIZATION: Expressing a number as a product of its prime factors | CUBE ROOT: The number that, when cubed, gives the original number
What's Next
What to Learn Next
Great job learning about cube numbers! Next, you should explore 'Cube Roots'. Understanding how to find cube roots will help you reverse the process of cubing numbers, which is super useful in solving many math problems and real-world scenarios!
