The third power of an integer: 0, 1, 8, 27, 64, 125... are perfect cubes.
A perfect cube is the result of multiplying an integer by itself three times. In other words, it is any number that can be written as n × n × n, or n³, where n is a whole number.
For example:
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1 × 1 × 1 = 1, so 1 is a perfect cube
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2 × 2 × 2 = 8, so 8 is a perfect cube
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3 × 3 × 3 = 27, so 27 is a perfect cube
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4 × 4 × 4 = 64, so 64 is a perfect cube
The name comes from geometry. If we build a cube with side length n, its volume is n³. A cube with side length 3 has a volume of 27 cubic units, which is why 27 is called a perfect cube.
Perfect cubes are closely related to perfect squares (numbers like 1, 4, 9, 16, 25 that result from multiplying an integer by itself twice). Both concepts are part of the broader study of powers and roots. Just as we can take the square root of a perfect square, we can take the cube root of a perfect cube: the cube root of 27 is 3.
When Do Students Learn About Perfect Cubes?
Students encounter perfect cubes after building familiarity with multiplication, exponents, and perfect squares.
Grades 4–5 – Introduction to Exponents and Powers
Students learn about squared numbers and begin exploring what it means to raise a number to a power, setting the stage for cubed numbers.
Grades 6–8 – Perfect Cubes and Cube Roots
Students identify perfect cubes, calculate cube roots, and work with cubic expressions in the context of volume and exponent rules.
Grades 9+ – Perfect Cubes in Algebra and Beyond
Students apply perfect cubes in algebraic factoring, including the sum and difference of cubes, and encounter them in polynomial and function work.
