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perfect cube — Definition, Formula & Examples

A perfect cube is a number you get when you multiply an integer by itself three times. For example, 27 is a perfect cube because 3×3×3=273 \times 3 \times 3 = 27.

An integer nn is a perfect cube if there exists an integer kk such that n=k3n = k^3. Perfect cubes can be positive, negative, or zero, since the cube of a negative integer is negative.

Key Formula

n=k3n = k^3
Where:
  • nn = The perfect cube
  • kk = An integer (positive, negative, or zero)

How It Works

To check whether a number is a perfect cube, try to find an integer whose cube equals that number. The first several positive perfect cubes are 1,8,27,64,125,216,343,512,729,10001, 8, 27, 64, 125, 216, 343, 512, 729, 1000. Unlike perfect squares, perfect cubes can be negative: (−2)3=−8(-2)^3 = -8, so −8-8 is a perfect cube. Taking the cube root reverses the operation — if n3\sqrt[3]{n} is an integer, then nn is a perfect cube.

Worked Example

Problem: Is 216 a perfect cube? If so, find the integer whose cube equals 216.
Try small cubes: List cubes of integers until you reach or pass 216.
53=125,63=2165^3 = 125,\quad 6^3 = 216
Verify: Multiply to confirm.
6×6×6=36×6=216  ✓6 \times 6 \times 6 = 36 \times 6 = 216 \; \checkmark
Answer: Yes, 216 is a perfect cube because 216=63216 = 6^3.

Visualization

Why It Matters

Recognizing perfect cubes helps you simplify cube roots and factor expressions like a3+b3a^3 + b^3 or a3−b3a^3 - b^3 in algebra. Volume problems in geometry also rely on perfect cubes — a cube with side length 5 cm has a volume of 125 cm3125\text{ cm}^3, and knowing that 125 is 535^3 lets you work backward from volume to side length.

Common Mistakes

Mistake: Thinking perfect cubes must be positive.
Correction: Negative integers have cubes too. For example, (−4)3=−64(-4)^3 = -64, so −64-64 is a perfect cube.

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