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Cubic and cube roots

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Cubic and Cube Roots

Cubing a number means multiplying it by itself three times; a cube root undoes that, asking what number cubed gives a result. Learn perfect cubes (like 27 = 3 cubed), why negative numbers have real cube roots unlike square roots, and worked examples.

What cubing means

Cubing a number means multiplying it by itself three times: 3 cubed (written 3³) is 3 × 3 × 3 = 27. The result of cubing a whole number is called a perfect cube — 27 is a perfect cube because it is 3³.

A cube root undoes cubing: it asks "what number, multiplied by itself three times, gives this result?" The cube root of 27 (written ∛27) is 3, because 3³ = 27. Cubing and cube-rooting are inverse operations, the same relationship squares and square roots have, just one power higher.

Cubing and cube roots are inverse operations Two arrows show inverse operations. Cubing: 3 cubed equals 27. Cube rooting: the cube root of 27 equals 3. Each operation undoes the other. 3³ cubing 27 27 cube root 3 ∛27 = 3 3³ = 27 Each undoes the other
Cubing and taking a cube root undo each other: 3³ = 27, and ∛27 = 3.

The first several perfect cubes are worth memorizing: 1³=1, 2³=8, 3³=27, 4³=64, 5³=125. Recognizing these makes their cube roots (1, 2, 3, 4, 5) instant to recall, and prime factorization helps confirm or simplify a cube root when the number isn't obviously a perfect cube.

Cube roots behave differently from square roots here: a negative number has no real square root, but it does have a real cube root. Since (−3) × (−3) × (−3) = −27, the cube root of −27 is −3. This is because multiplying three negatives gives a negative result, while multiplying two negatives gives a positive one.

  • Cubing: 4³ = 4 × 4 × 4 = 64.
  • Cube root: ∛64 = 4, since 4³ = 64.
  • Negative cube root: ∛−8 = −2, since (−2)³ = −8.

The same radical family extends to multiplying and dividing radicals, which covers square roots, cube roots, and beyond together.

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