Is 51 a Perfect Cube? No

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 51 Is Not a Perfect Cube

1. Try ∛51 ≈ 3.7084  (not an integer).
2. Prime factorization of |n| = 51:
   51 = 3 × 17
3. The exponent of 3 is 1 — not divisible by 3. A number is a perfect cube if and only if every prime in its factorization has an exponent divisible by 3, so 51 cannot be written as k³.
4. Nearest perfect cubes:
   • 27 = 3³  (just below)
   • 64 = 4³  (just above)

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 51 — decimal value with Newton's Method
• What is 51 Cubed? — compute n³
• Simplify ∛51 — radical form (e.g. 2∛9)

References

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

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