Is 1 a Perfect Cube? Yes — ∛1 = 1

Perfect Cube Checker

Use the checker above to determine whether any integer is a perfect cube. A perfect cube is an integer of the form k³ for some integer k. Unlike perfect squares, perfect cubes can be negative: since (-k)³ = -k³, every negative integer whose absolute value is a perfect cube is itself a perfect cube. The first non-negative perfect cubes are 0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, ...

Step-by-Step: Why 1 Is a Perfect Cube

1. Compute ∛1:
   ∛1 = 1 (an integer).
2. Verify by cubing:
   1 × 1 × 1 = 1 ✓
3. By convention, 1 = 1³ is the trivial perfect cube.

∛1 = 1

What Is a Perfect Cube?

A perfect cube is an integer that is the cube of an integer:

n is a perfect cube  ⇔  n = k³,  k ∈ ℤ

Equivalently, n is a perfect cube if and only if ∛n is an integer. Geometrically, you can arrange n identical unit cubes into a cubic grid only if n is a non-negative perfect cube.

Signs: Unlike perfect squares (which are always non-negative), perfect cubes can be negative. Examples of negative perfect cubes: -1 = (-1)³, -8 = (-2)³, -27 = (-3)³, -64 = (-4)³, -125 = (-5)³.

Prime-factorization rule: |n| is a perfect cube iff every prime in its prime factorization appears with an exponent divisible by 3. For example, 216 = 2³ × 3³ (both divisible by 3) is a perfect cube; 72 = 2³ × 3² has exponent 2 on 3, so it is not.

Nearby Examples

Related Operations

• Cube Root of 1 — decimal value with Newton's Method
• What is 1 Cubed? — compute n³

References

• [1] Cube (algebra) (Wikipedia)
• [2] Cubic Number (Wolfram MathWorld)

Popular Perfect-Cube Checks

Common ‘Is N a perfect cube?’ queries — click any to see the step-by-step verdict (includes negative cubes):