Perfect Cube Calculator & Checker
Check if any number is a perfect cube
Quick Test
Enter any integer (positive or negative) to check if it equals a whole number multiplied by itself 3 times (k³ = n).
Perfect Cube Calculator
Check whether any number (positive or negative) is a perfect cube instantly. Get base integer roots, prime factor powers, nearest bounding cubes, and algebraic proofs. 100% free.
Instant Proof Bounding Cubes Prime Powers 100% Free
Table of Contents 10 Sections Show Hide
Calculator Guide
How to Use the Perfect Cube Calculator
Our interactive tool checks numbers of any magnitude in milliseconds. Follow these 3 easy steps:
1
Enter Any Integer
Type any positive or negative whole number into the input box, or click one of the preset chips like -27, 216, or 1331.
2
Click 'Check Perfect Cube'
Click the check button or hit Enter. The tool immediately computes the exact real cube root and tests integer divisibility.
3
Review the Full Proof
Get an instant verdict banner, base root, prime factor powers, nearest bounding cubes, and the exact multiplier needed if it is not a cube.
Definition
What Is a Perfect Cube?
A perfect cube (or cube number) is an integer obtained by multiplying another whole number by itself three times: n = k³ = k × k × k.
For example, 27 is a perfect cube because 3 × 3 × 3 = 27. Similarly, 125 is a perfect cube because 5 × 5 × 5 = 125. In contrast, 50 is not a perfect cube because its cube root is approximately 3.684, which is not an integer.
The term cube comes directly from 3D geometry: if you have 27 unit blocks, you can build a solid 3×3×3 geometric cube with equal length, width, and height. With 50 blocks, you can never assemble a solid cube without leftover pieces.
3³ = 27 ∛27 = 3 (Integer Root)
5³ = 125 ∛125 = 5 (Integer Root)
(-4)³ = -64 ∛(-64) = -4 (Real Root!)
Negative Cubes Exist!
Unlike square roots, odd roots preserve signs! (-3)³ = (-3) × (-3) × (-3) = -27. Therefore, negative integers can be legitimate real perfect cubes.
Zero & One
0 and 1 are trivial perfect cubes: 0³ = 0 and 1³ = 1. Negative one is also a cube: (-1)³ = -1.
Parity Consistency
The cube of an even integer is always even (2³=8, 4³=64), and the cube of an odd integer is always odd (3³=27, 5³=125).
Cube of Factors
In prime factorization, an integer is a perfect cube if and only if the exponent of every prime factor is a multiple of 3.
Mathematical Proofs
4 Proven Tests to Check Any Perfect Cube
Mathematicians use four rigorous tests to verify whether an unknown integer n is a perfect cube:
If √‾N‾ = k ∈ ℤ, then N is a Perfect Cube (k × k × k = N)
1
Test 1: The Integer Cube Root Test
Direct Computational CheckTake the cube root ∛n. If the result is an exact integer with zero fractional remainder, n is a perfect cube.
∛343 = 7.0000000 → Integer 7 confirmed.
2
Test 2: The Prime Factorization Rule
All Exponents Must Be Multiples of 3Break n down into its prime factors. If all prime exponents are divisible by 3 (3, 6, 9...), then n is a perfect cube.
216 = 2³ × 3³ (exponents are 3 and 3, both divisible by 3) → Perfect cube!
3
Test 3: Negative Cubes & Symmetry Rule
Odd Powers Preserve the Negative SignBecause (-k)³ = -k³, a negative number -n is a perfect cube if and only if |n| is a perfect cube.
-512 is a perfect cube because 512 = 8³, making ∛(-512) = -8.
4
Test 4: The Digital Root (Mod 9) Filter
Quick Elimination TrickSum the digits of n repeatedly until you get a single digit. The digital root of any perfect cube is ALWAYS 0, 1, 8, or 9 (in mod 9 arithmetic: 0, ±1). If the digital root is 2, 3, 4, 5, 6, or 7, it is guaranteed NOT to be a cube!
Digital root of 1331: 1+3+3+1 = 8 (passes test).
Step-by-Step Proofs
Step-by-Step Worked Verification Examples
PERFECT CUBE
Example 1: Testing 512
∛512 = 8 : 8 × 8 × 8 = 512. Prime factorization is 2⁹ with exponent 9 divisible by 3. Digital root: 5 + 1 + 2 = 8 (valid).
NOT A CUBE
Example 2: Testing 100
∛100 ≈ 4.642 : 100 sits strictly between 4³ = 64 and 5³ = 125. Prime factors are 2² × 5², neither exponent is divisible by 3.
REAL NEGATIVE CUBE
Example 3: Testing -216
∛(-216) = -6 : (-6) × (-6) × (-6) = 36 × (-6) = -216. Odd roots preserve negative signs, producing a valid real integer root.
Mental Math
Mental Math Shortcuts: The 1-to-1 Last-Digit Bijection
Unlike perfect squares (which can only end in 0, 1, 4, 5, 6, or 9), every digit from 0 to 9 can be the last digit of a perfect cube. In fact, there is a strict one-to-one bijection between the ones digit of a base integer and the ones digit of its cube:
| Base Last Digit (k) | Cube Calculation (k³ mod 10) | Cube Last Digit (k³) | Memory Rule |
|---|---|---|---|
| 0 | 0³ = 0 | 0 | Preserved (Same digit) |
| 1 | 1³ = 1 | 1 | Preserved (Same digit) |
| 2 | 2³ = 8 | 8 | Complements to 10 (2 + 8 = 10) |
| 3 | 3³ = 27 | 7 | Complements to 10 (3 + 7 = 10) |
| 4 | 4³ = 64 | 4 | Preserved (Same digit) |
| 5 | 5³ = 125 | 5 | Preserved (Same digit) |
| 6 | 6³ = 216 | 6 | Preserved (Same digit) |
| 7 | 7³ = 343 | 3 | Complements to 10 (7 + 3 = 10) |
| 8 | 8³ = 512 | 2 | Complements to 10 (8 + 2 = 10) |
| 9 | 9³ = 729 | 9 | Preserved (Same digit) |
PRO TRICK
Extracting Cube Roots of Large Numbers (Up to 1,000,000) in 3 Seconds
To find the cube root of 175,616:
1. Ones Digit: The number ends in 6. Looking at our table, a cube ending in 6 must have a root ending in 6.
2. Tens Digit: Cross off the last three digits (616), leaving 175. The largest perfect cube below 175 is 125 (5³). So the tens digit is 5.
Result: ∛175616 = 56. Verified: 56 × 56 × 56 = 175,616!
Special Cases
Edge Cases: Zero, Negative Cubes, Fractions, and Decimals
Odd powers behave differently from even powers. Here is how special numeric sets are evaluated:
1. Is Zero a Perfect Cube?
Yes. 0 is an integer, and 0 × 0 × 0 = 0 (0³ = 0). Its principal cube root is exactly 0.
2. Negative Perfect Cubes
Fully Real. Unlike square roots which require the imaginary unit i, odd roots preserve signs: (-k)³ = -k³. For example, ∛(-64) = -4.
3. Rational Fractions
A fraction is a cubic power if both its numerator and denominator are perfect cubes: 27/64 = (3/4)³.
4. Finite Decimals
Decimals are cubes if the significant digits form a cube and the number of decimal places is a multiple of 3: 0.027 = (0.3)³, 0.000125 = (0.05)³.
Comparing cubic numbers with square numbers? Test whether an integer is a perfect square →
Bounding Intervals
Finding the Nearest Perfect Cube for Non-Cube Numbers
When a number is not a cube, it always sits between two consecutive integer cubes: k³ < n < (k + 1)³.
For example, 500 is not a cube. The nearest cubes are 7³ = 343 and 8³ = 512. The distance to 343 is 157, while the distance to 512 is only 12. Therefore, 512 is the closest cube to 500.
| Target (N) | Cube Root (∛N) | Lower Cube (k³) | Upper Cube ((k+1)³) | Nearest Cube | Distance |
|---|---|---|---|---|---|
| 50 | 3.684 | 27 (3³) | 64 (4³) | 64 | 64 - 50 = 14 |
| 200 | 5.848 | 125 (5³) | 216 (6³) | 216 | 216 - 200 = 16 |
| 500 | 7.937 | 343 (7³) | 512 (8³) | 512 | 512 - 500 = 12 |
| 800 | 9.283 | 729 (9³) | 1000 (10³) | 729 | 800 - 729 = 71 |
When an integer is not a perfect cube, its root is an irrational decimal. Compute exact decimal cube roots with radical steps →
Reference Table
The 10 Perfect Cubes from 1 to 1000 Reference Table
Between 1 and 1000 inclusive, there are exactly 10 perfect cubes:
| Base Root (k) | Formula (k × k × k) | Perfect Cube (k³) | Prime Factorization | ∛k³ |
|---|---|---|---|---|
| 1 | 1 × 1 × 1 | 1 | 1 | 1 |
| 2 | 2 × 2 × 2 | 8 | 2³ | 2 |
| 3 | 3 × 3 × 3 | 27 | 3³ | 3 |
| 4 | 4 × 4 × 4 | 64 | 2⁶ = (2²)³ | 4 |
| 5 | 5 × 5 × 5 | 125 | 5³ | 5 |
| 6 | 6 × 6 × 6 | 216 | 2³ × 3³ | 6 |
| 7 | 7 × 7 × 7 | 343 | 7³ | 7 |
| 8 | 8 × 8 × 8 | 512 | 2⁹ = (2³)³ | 8 |
| 9 | 9 × 9 × 9 | 729 | 3⁶ = (3²)³ | 9 |
| 10 | 10 × 10 × 10 | 1000 | 2³ × 5³ | 10 |
1 - 10,000
Extended List of Perfect Cubes from 1 to 10,000
Between 1 and 10,000 inclusive, there are exactly 21 perfect cubes because 21³ = 9,261 and 22³ = 10,648.
| k | k³ | k | k³ | k | k³ |
|---|---|---|---|---|---|
| 1 | 1 | 8 | 512 | 15 | 3375 |
| 2 | 8 | 9 | 729 | 16 | 4096 |
| 3 | 27 | 10 | 1000 | 17 | 4913 |
| 4 | 64 | 11 | 1331 | 18 | 5832 |
| 5 | 125 | 12 | 1728 | 19 | 6859 |
| 6 | 216 | 13 | 2197 | 20 | 8000 |
| 7 | 343 | 14 | 2744 | 21 | 9261 |
Evaluating higher powers beyond the 3rd degree? Evaluate arbitrary nth degree radical roots →
Code & Spreadsheets
How to Check for Perfect Cubes in Code and Spreadsheets
Verify cube numbers automatically in Excel, Google Sheets, Python, and JavaScript:
1. Microsoft Excel & Google Sheets
=IF(ROUND(A2^(1/3), 0)^3 = A2, "Perfect Cube", "Not a Cube") 2. Python (Exact Integer Logic)
def is_cube(n): return round(abs(n)**(1/3))**3 == abs(n) 3. JavaScript
const isCube = n => Math.round(Math.cbrt(n))**3 === n;
Curriculums
International Curriculum Standards: US, UK, Canada, Australia, Germany, and France
US, UK, Canada & Australia
Terminology: Perfect Cube or Cube Number.
- United States: Common Core standard CCSS.MATH.CONTENT.8.EE.A.2 (Grade 8).
- United Kingdom: GCSE Mathematics Subject Content 8.01 (Edexcel, AQA, OCR non-calculator paper).
- Canada & Australia: Australian curriculum ACMNA180 and Canadian provincial Grade 8/9 units.
Germany & France
Terminology: Kubikzahlen (German) and Cubes parfaits (French).
- Germany (DIN 1333): Taught in Realschule/Gymnasium (Klassen 8/9) with radical exactitude.
- France (Collège Cycle 4): Classes de 4ème/3ème under formal power rules and volume computations.
- Uses decimal comma notation (∛50 ≈ 3,684).
Practical Uses
Real-World Applications of Perfect Cubes
Volume & Packaging
Calculating box volumes, container dimensions, and material density in logistics and shipping.
Physics & Fluid Dynamics
Determining mass and buoyancy scaling according to the square-cube law in biomechanics and aviation.
Computer Graphics & Voxels
3D voxel grid calculations, octree spatial indexing, and Minecraft-style block coordinate structures.
Number Theory & Cryptography
Elliptic curve cryptography and Diophantine equations involving sums of cubes (Fermat's Last Theorem).
Mistakes to Avoid
Common Mistakes and Misconceptions to Avoid
#1
Confusing Cubing with Multiplying by 3
Cubing a number means multiplying it by itself three times (4³ = 4 × 4 × 4 = 64), not multiplying it by three (4 × 3 = 12).
#2
Assuming Negative Numbers Cannot Be Cubes
Unlike square roots, odd roots preserve negative signs. Multiplying three negative numbers gives a negative product: (-5) × (-5) × (-5) = -125.
#3
Trying to Rule Out Cubes by Last Digits Alone
Every digit from 0 to 9 can end a perfect cube. You cannot eliminate a candidate by its last digit alone; you must use prime factors or modulo 9 digital roots.
#4
Floating-Point Rounding Pitfalls in Calculators
In computers, computing 125^(1/3) frequently yields 4.999999999999999. Naively checking if int(125^(1/3)) == 5 can fail without proper rounding logic.
Questions & Answers
Frequently Asked Questions
What is a perfect cube in simple terms?
A perfect cube is any whole number that you can get by multiplying another whole number by itself three times. For example, 27 is a perfect cube because 3 × 3 × 3 = 27.
Can a negative number be a perfect cube?
Yes! Multiplying an odd number of negative factors results in a negative product: (-4) × (-4) × (-4) = -64. Thus, -64 is a valid real perfect cube with cube root -4.
Is 0 considered a perfect cube?
Yes. Zero is an integer, and 0 × 0 × 0 = 0 (0³ = 0). Its principal cube root is 0.
How do you check if a large number is a perfect cube without a calculator?
Find the prime factorization of the number. If every prime factor has an exponent divisible by 3, the number is a perfect cube. In addition, compute its digital root: if the repeated digit sum is not 1, 8, or 9, it cannot be a cube.
Why is 64 both a perfect square and a perfect cube?
A number that is both a square and a cube is a sixth power (k⁶). Because 2⁶ = 64, it can be written as 8² = 64 and 4³ = 64. Numbers of this form are known as perfect sixth powers.
What is the smallest positive perfect cube?
The smallest positive perfect cube is 1, because 1 × 1 × 1 = 1. If non-positive integers are included, 0 is smaller, and negative cubes extend infinitely to negative infinity.
How many perfect cubes are there between 1 and 100?
There are exactly 4 perfect cubes between 1 and 100 inclusive: 1 (1³), 8 (2³), 27 (3³), and 64 (4³). The next cube is 125 (5³), which exceeds 100.
Is 100 a perfect cube?
No. The cube root of 100 is approximately 4.6416, which is not an integer. The prime factorization of 100 is 2² × 5², neither of which has an exponent divisible by 3.
What is the difference between a square root and a cube root?
A square root asks what number multiplied by itself gives the original value. A cube root asks what number multiplied by itself three times gives the original value.
Is this online perfect cube calculator free?
Yes. Our calculator is 100% free with unlimited checks, instant answers, and detailed step-by-step mathematical proofs.
Summary
A Quick Recap
Three Equal Factors
A perfect cube satisfies n = k³, meaning three identical integers multiply to give n.
Negative Friendly
Unlike square roots, cube roots of negative numbers are real and valid integers.
Prime Factor Exponents
All prime exponents must divide cleanly by 3 in prime factorization.
100% Free Tool
Use our interactive checker above anytime for instant proofs, bounds, and steps.