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A cube root is the number that, when multiplied by itself twice, gives the original number. For a cube in geometry, each side length (ℓ) is the cube root of the volume (V), because V = ℓ³.
Mathematically, we write this as ℓ = ³√V.
For whole numbers between 1 and 100, memorizing the cubes of 1–10 is a handy shortcut. The table below lists the results:
| 1³ | 1 |
| 2³ | 8 |
| 3³ | 27 |
| 4³ | 64 |
| 5³ | 125 |
| 6³ | 216 |
| 7³ | 343 |
| 8³ | 512 |
| 9³ | 729 |
| 10³ | 1,000 |
With this table in mind, you can quickly identify the integer cube root of any number in that range.
When the number is not a perfect cube, the most reliable approach is estimation followed by refinement. Start by bracketing the target between two consecutive cubes. Then adjust your guess and cube it again until the result is sufficiently close.
Since 1³ = 1 and 2³ = 8, ³√3 lies between 1 and 2. A quick trial gives 1.5³ = 3.375 (too high) and 1.4³ = 2.744 (too low). The precise value, accurate to six decimal places, is 1.442249. Because it is irrational, no exact integer will satisfy the equation.
Factor 81 as 3 × 3 × 3 × 3. The first three 3’s cancel with the cube root, leaving 3 × ³√3. Using the value from above:
³√81 = 3 × 1.442249 = 4.326747.
1. ³√150
Between 125 (5³) and 216 (6³). Trial values: 5.3³ = 148.88 (too low), 5.4³ = 157.46 (too high). Refining further yields 5.313293.
2. ³√1,029
Factor 1,029 = 7 × 7 × 7 × 3. Thus ³√1,029 = 7 × ³√3 = 10.095743.
3. ³√(–27)
Cube roots of negative numbers remain negative, so ³√(–27) = –3.
