How to Check if a Number is a Perfect Cube?
The concept of Perfect Cube Of Numbers plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding perfect cubes helps students to solve cube root problems, algebraic identities, and pattern-based questions easily in classes as well as competitive exams.
What Is Perfect Cube Of Numbers?
A perfect cube is defined as a number that results from multiplying an integer by itself three times, or, in other words, raising a whole number to the power of three. For example, the number 27 is a perfect cube because 3 × 3 × 3 = 27. You’ll find this concept applied in areas such as number systems, cube roots, and algebraic identities.
Key Formula for Perfect Cube Of Numbers
Here’s the standard formula: \( n^3 = n \times n \times n \), where n is a whole number. If the result is an integer, it is called a perfect cube. For example, \( 5^3 = 5 \times 5 \times 5 = 125 \).
Perfect Cube Number Table (1 to 20, 1 to 30, 1 to 50, 1 to 100)
Memorizing the most common perfect cubes can speed up exam calculations. Below is a compact table of perfect cube numbers from 1 to 20. You can easily make a similar chart up to 100 or even 1000 for fast revision.
| Number (n) | Perfect Cube (n³) |
|---|---|
| 1 | 1 |
| 2 | 8 |
| 3 | 27 |
| 4 | 64 |
| 5 | 125 |
| 6 | 216 |
| 7 | 343 |
| 8 | 512 |
| 9 | 729 |
| 10 | 1000 |
| 11 | 1331 |
| 12 | 1728 |
| 13 | 2197 |
| 14 | 2744 |
| 15 | 3375 |
| 16 | 4096 |
| 17 | 4913 |
| 18 | 5832 |
| 19 | 6859 |
| 20 | 8000 |
Properties of Perfect Cube Numbers
- The cube of an even number is always even. (e.g. 4³ = 64)
- The cube of an odd number is always odd. (e.g. 5³ = 125)
- Perfect cubes may be positive or negative (e.g. (-3)³ = -27).
- No perfect cube ends with the digits 2, 3, 7, or 8.
- A perfect cube can be obtained only by multiplying a number three times by itself (n × n × n).
- Sum of certain consecutive odd numbers results in a perfect cube.
How to Check if a Number is a Perfect Cube
Here is a clear stepwise method used in maths exams and Olympiads:
- Prime Factorization: Express the number as the product of prime factors.
- Group the factors in triples. Each prime must appear a multiple of 3 times.
- If every factor can be grouped with no "leftovers," the number is a perfect cube.
Example: Is 216 a perfect cube?
2. Group: (2×2×2) × (3×3×3)
3. Both groups have three of the same factor. So, 216 is a perfect cube.
4. Cube root: 2 × 3 = 6, so 216 = 6³.
Speed Trick or Vedic Shortcut
A quick trick for checking big perfect cubes: Find the sum of the digits and check if the result is 1, 8, or 9 (keep adding digits until you get one number). This is not a foolproof method but helps skip non-cube candidates fast.
Example: Check if 512 is a perfect cube.
5 + 1 + 2 = 8 (which fits the shortcut), so check by factorization. 512 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 (nine 2s). Group as (2×2×2) × (2×2×2) × (2×2×2). Yes, 512 is a perfect cube: 512 = 8³.
Try These Yourself
- Write the first five perfect cubes of numbers.
- Check if 128 is a perfect cube.
- Find all perfect cubes between 30 and 100.
- Identify non-perfect cubes from the list: 64, 100, 125.
Step-by-Step Illustration
Question: What is the smallest number by which 1323 should be multiplied to make it a perfect cube?
2. Groups: (3×3×3) forms one group. Two sevens are left (we need three for a cube).
3. Multiply by another 7 → 1323 × 7 = 9261
4. 9261 = (3×3×3) × (7×7×7), is a perfect cube.
Answer: 1323 should be multiplied by 7.
Frequent Errors and Misunderstandings
- Thinking a number is a perfect cube just because the digit sum matches 1, 8, or 9 (always check with factorization too).
- Confusing cubes and squares (cube = three times, square = two times).
- Ignoring negative numbers—negative integers can also have perfect cubes.
Relation to Other Concepts
The idea of Perfect Cube Of Numbers connects closely with cube roots, prime factorization, and special algebraic identities like the cube of the sum or difference of binomials. Mastering cubes will help in topics like cube root of numbers or working with square numbers as well.
Cross-Disciplinary Usage
Perfect cube numbers are not only useful in Maths but also play an important role in Physics, Computer Science, and logical reasoning. For instance, calculating the volume of a cube in geometry, understanding data encoding in computer science, and solving algebraic sums in Olympiads. Students preparing for JEE or NEET will see its relevance in various questions across subjects.
Classroom Tip
A quick way to remember perfect cubes up to 10 or 20 is to recite the cube table every day. Stick the cube number chart on your study table or use printable charts available on Vedantu for daily revision. Vedantu’s live teachers also use quick cube tricks during practice classes for memory retention.
Strong Internal Links for Further Practice
- Cube Root of Numbers – Learn to find cube roots and practice reverse calculation.
- Prime Factorization – Master the skill to check if a number is a perfect cube.
- Cube Root Table – Get a ready-made look-up chart for MCQs and quick revision.
- Cube of the Difference of Two Binomials – Apply your perfect cube understanding to algebra.
We explored Perfect Cube Of Numbers—from definition, formula, example tables, error checking, and links to other maths topics. Revise your cubes and keep practicing on Vedantu to become confident in exams and competitions!
FAQs on Perfect Cube of Numbers: Definition, Formula, Table & FAQs
1. What is a perfect cube in mathematics?
A perfect cube is a number obtained by multiplying an integer by itself three times (n × n × n). It's the result of cubing an integer. For example, 27 is a perfect cube because 3 × 3 × 3 = 27.
2. What is the formula for a perfect cube?
The perfect cube formula is simply n³ = n × n × n, where 'n' is any integer. This means to find the perfect cube of a number, you multiply that number by itself three times.
3. How do I check if a number is a perfect cube?
One way to check is through prime factorization. Find the prime factors of the number. If all prime factors can be grouped into sets of three, the number is a perfect cube. Another method is to find its cube root; if the cube root is a whole number, then the original number is a perfect cube.
4. What are the first 10 perfect cubes?
The first ten perfect cubes are: 1, 8, 27, 64, 125, 216, 343, 512, 729, and 1000.
5. What is the difference between a perfect square and a perfect cube?
A perfect square is obtained by multiplying an integer by itself (n × n), while a perfect cube is obtained by multiplying an integer by itself three times (n × n × n).
6. Can negative numbers be perfect cubes?
Yes, negative numbers can also be perfect cubes. For example, -8 is a perfect cube because (-2) × (-2) × (-2) = -8.
7. How do I find the smallest number to multiply by to make a number a perfect cube?
Use prime factorization. Find the prime factors, group them in threes, and identify which factors are missing to complete a set of three. The product of these missing factors is the smallest number you need to multiply by.
8. What are some properties of perfect cubes?
• The cube of an even number is always even.
• The cube of an odd number is always odd.
• Perfect cubes can end in 0, 1, 4, 5, 6, or 9.
• A perfect cube can be expressed as the sum of consecutive odd numbers.
9. What is the cube root of a perfect cube?
The cube root of a perfect cube is the integer that, when multiplied by itself three times, gives the perfect cube. For example, the cube root of 64 is 4 (because 4 × 4 × 4 = 64).
10. How are perfect cubes used in algebra?
Perfect cubes are used in various algebraic manipulations, including factoring cubic expressions using the sum or difference of cubes formulas: a³ + b³ = (a + b)(a² - ab + b²) and a³ - b³ = (a - b)(a² + ab + b²).
11. Are there any quick ways to identify perfect cubes?
While prime factorization is reliable, checking the last digit can sometimes help eliminate options. Perfect cubes can only end in 0, 1, 8, 7, 4, or 5. However, this is not a definitive test, and prime factorization provides conclusive confirmation.
12. Why is it helpful to memorize some perfect cubes?
Memorizing perfect cubes (especially smaller ones) significantly speeds up calculations involving cube roots and simplifies solving problems in algebra and other areas of mathematics. It is a valuable tool for competitive exams.
