No, cube roots are sometimes irrational numbers, but not always.
Just like with square roots, some cube roots result in irrational numbers, while others result in rational numbers. Whether a cube root is rational or irrational depends on whether the number under the cube root symbol is a perfect cube.
Rational vs. Irrational Cube Roots
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Rational Cube Roots: A cube root is rational if the number under the cube root symbol is a perfect cube (i.e., a number that can be obtained by cubing an integer). For example, the cube root of 8 is 2 (∛8 = 2), and the cube root of 27 is 3 (∛27 = 3). 8 and 27 are perfect cubes (2³ = 8 and 3³ = 27), and their cube roots are rational numbers.
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Irrational Cube Roots: A cube root is irrational if the number under the cube root symbol is not a perfect cube. For example, the cube root of 2 (∛2) is an irrational number. There is no integer or fraction that, when multiplied by itself three times, equals 2. Similarly, ∛3, ∛4, ∛5, ∛6, ∛7, ∛9, ∛10, etc., are all irrational numbers.
Examples:
Here's a table summarizing rational and irrational cube roots:
| Number | Cube Root | Rational/Irrational | Explanation |
|---|---|---|---|
| 1 | ∛1 = 1 | Rational | 1 is a perfect cube (1³ = 1) |
| 8 | ∛8 = 2 | Rational | 8 is a perfect cube (2³ = 8) |
| 27 | ∛27 = 3 | Rational | 27 is a perfect cube (3³ = 27) |
| 64 | ∛64 = 4 | Rational | 64 is a perfect cube (4³ = 64) |
| 2 | ∛2 ≈ 1.260 | Irrational | 2 is not a perfect cube |
| 3 | ∛3 ≈ 1.442 | Irrational | 3 is not a perfect cube |
| 5 | ∛5 ≈ 1.710 | Irrational | 5 is not a perfect cube |
Key Takeaway
In conclusion, only the cube roots of perfect cubes are rational numbers. The cube roots of all other numbers are irrational.
