Input
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Result
Enter values and calculate to see exact and decimal forms.
Example data
Use these examples to validate outputs.
| Numerator | Denominator | Exact (unrationalized) | Decimal (6 places) |
|---|---|---|---|
| 8 | 27 | 2/3·∛(1/1) | 0.666667 |
| 16 | 54 | 2/3·∛(1/1) | 0.666667 |
| -125 | 64 | -5/4·∛(1/1) | -1.250000 |
| 2 | 3 | 1·∛(2/3) | 0.873580 |
| 1 | 8 | 1/2·∛(1/1) | 0.500000 |
Formula used
For \( a,b \in \mathbb{Z}, b\ne0 \): \( \sqrt[3]{\tfrac{a}{b}} = \tfrac{\sqrt[3]{a}}{\sqrt[3]{b}} \). Extract perfect cubes: \( a = k_a^3 r_a, \; b = k_b^3 r_b \) with \( r_a, r_b \) cube‑free. Then \( \sqrt[3]{\tfrac{a}{b}} = \tfrac{k_a}{k_b}\sqrt[3]{\tfrac{r_a}{r_b}} \). To rationalize: multiply numerator and denominator by \( \sqrt[3]{r_b^2} \) giving \( \tfrac{k_a}{k_b r_b}\sqrt[3]{r_a r_b^2} \).
How to use this calculator
- Enter numerator and denominator. Negative values are supported.
- Choose desired decimal precision.
- Optionally enable “Show rationalized form” and “Show algebraic steps”.
- Press Calculate to see exact and decimal outputs.
- Export the result or example table using CSV or PDF buttons.
FAQs
Perfect‑cube fractions & exact roots (data)
Fractions whose numerator and denominator are perfect cubes. Exact roots are rational.
| Fraction | ∛(numerator) | ∛(denominator) | Exact root | Decimal (6 places) |
|---|---|---|---|---|
| 1/1 | 1 | 1 | 1 | 1.000000 |
| 8/27 | 2 | 3 | 2/3 | 0.666667 |
| 27/64 | 3 | 4 | 3/4 | 0.750000 |
| 64/125 | 4 | 5 | 4/5 | 0.800000 |
| 125/216 | 5 | 6 | 5/6 | 0.833333 |
| 216/343 | 6 | 7 | 6/7 | 0.857143 |
| 343/512 | 7 | 8 | 7/8 | 0.875000 |
| 1000/8000 | 1 | 2 | 1/2 | 0.500000 |
Simplification patterns for sample fractions (data)
Examples showing largest perfect‑cube factors, unrationalized and rationalized forms.
| Input | Reduced | Factorization | Unrationalized exact | Rationalized exact | Decimal (6 places) |
|---|---|---|---|---|---|
| 16/54 | 8/27 | |8| = 2^3 × 1, 27 = 3^3 × 1 | 2/3·∛(1/1) | 2/3·∛(1) | 0.666667 |
| 250/108 | 125/54 | |125| = 5^3 × 1, 54 = 3^3 × 2 | 5/3·∛(1/2) | 5/6·∛(4) | 1.322834 |
| -72/40 | -9/5 | |-9| = 1^3 × 9, 5 = 1^3 × 5 | -1·∛(9/5) | -1/5·∛(225) | -1.216440 |
| 2/3 | 2/3 | |2| = 1^3 × 2, 3 = 1^3 × 3 | 1·∛(2/3) | 1/3·∛(18) | 0.873580 |
| 18/20 | 9/10 | |9| = 1^3 × 9, 10 = 1^3 × 10 | 1·∛(9/10) | 1/10·∛(900) | 0.965489 |
| 54/16 | 27/8 | |27| = 3^3 × 1, 8 = 2^3 × 1 | 3/2·∛(1/1) | 3/2·∛(1) | 1.500000 |