A fraction raised to a fractional power, such as (8/27)^(2/3), combines three ideas: powers of fractions, roots, and the meaning of a rational exponent. This calculator works through all three, gives the exact fraction when one exists, and otherwise shows the radical form and a decimal approximation.
How to use the calculator
- Enter the base as a fraction (or mixed number).
- Enter the exponent as a fraction: numerator = power, denominator = root. For a whole-number exponent use a denominator of 1; for a negative exponent put the minus sign on its numerator.
- The tape shows the decimal value, the exact fraction when there is one, and the root and power. The steps explain each rule as it is applied.
The rules
- Reduce the exponent to lowest terms: 6/8 becomes 3/4.
- Negative exponent: flip the base and make the exponent positive.
- Root: take the n-th root of the numerator and the denominator.
- Power: raise the result to the m-th power.
Taking the root first keeps the numbers small. You can also raise to the power first and take the root afterward — the answer is the same for positive bases.
Worked example: (8/27)^(2/3)
- The exponent 2/3 means “cube root, then square.”
- Cube roots: ∛8 = 2 and ∛27 = 3, so the root is 2/3.
- Square it: (2/3)² = 4/9 ≈ 0.4444.
With a negative exponent: (4/9)^(−3/2). Flip to (9/4)^(3/2). Square roots: √9 = 3 and √4 = 2, giving 3/2. Cube it: (3/2)³ = 27/8 = 3 3/8.
Special cases
| Case | Result | Example |
|---|---|---|
| Exponent 0 | 1 (base not 0) | (3/4)^0 = 1 |
| Exponent 1/2 | Square root | (4/9)^(1/2) = 2/3 |
| Whole-number exponent | Power of top and bottom | (3/2)^5 = 243/32 |
| Negative base, odd root | Negative result | (−8/27)^(1/3) = −2/3 |
| Negative base, even root | Not a real number | (−4/9)^(1/2) — undefined |
| Base 0, negative exponent | Undefined | 0^(−1/2) — division by 0 |
Irrational results
When the parts are not perfect powers, the answer is irrational. (2/3)^(1/2) = √2/√3. Rationalizing the denominator gives √6/3 ≈ 0.8165. The calculator shows the radical form and the decimal; to simplify radicals further, use the simplify radicals calculator.
Reducing the exponent first
For a negative base, writing the exponent in lowest terms matters. (−8)^(2/6) reduces to (−8)^(1/3) = −2, which is the convention most textbooks and this calculator follow. Computing (−8)² = 64 first and then taking a sixth root would give 2 instead, which is why exponents are always reduced before working with negative bases.
For whole-number bases and powers, the exponent calculator and the nth root calculator cover the same rules in more depth.
Frequently asked questions
What does a fractional exponent mean?
The denominator of the exponent is a root and the numerator is a power. x^(2/3) means the cube root of x, squared. So 8^(2/3) = (∛8)² = 2² = 4.
How do you raise a fraction to a power?
Apply the power to the numerator and the denominator separately: (a/b)^n = aⁿ/bⁿ. With a fractional exponent, take the root of each part first: (8/27)^(2/3) = (∛8/∛27)² = (2/3)² = 4/9.
What does a negative exponent do to a fraction?
It flips the fraction. (a/b)^(−n) = (b/a)^n. For example (4/9)^(−3/2) = (9/4)^(3/2) = (3/2)³ = 27/8.
Why do I get an error for a negative base?
An even root of a negative number is not a real number — no real number squared gives −4/9. Odd roots are fine: (−8/27)^(1/3) = −2/3.
Why is the answer sometimes only a decimal?
If the numerator or denominator is not a perfect power for the root, the result is irrational. (2/3)^(1/2) = √2/√3 ≈ 0.8165 cannot be written as a fraction of whole numbers, so it is shown in radical form and as a rounded decimal.