Math
Fractional Exponent Calculator
Raise any whole number, fraction, or decimal to a fractional exponent m/n.
The calculator reduces the exponent, takes the nth root, raises to the mth power, and reports an exact fraction when the base is a perfect power, a simplified radical when it is not, and always the decimal value. Negative exponents and negative bases with odd roots are handled.
Type a fraction (3/4), a mixed number (2 1/3), a whole number, or a decimal (1.75). Repeating decimals work too: 0.(3).
Use a negative number for a negative exponent, such as -2.
The root index. 2 = square root, 3 = cube root.
Try an example
Result
Exact value
—
- Decimal
- —
- Radical form
- —
- Reduced exponent
- —
- Rational answer?
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Student quick launch
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Study path
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Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
What a fractional exponent means
An The small raised number saying how many times to multiply the base by itself. of m/n means take the nth root and raise to the mth power, in either order. The The bottom number of a fraction; it says how many equal parts make one whole. is the root and the The top number of a fraction; it counts how many parts you have. is the power. 8^(2/3) is the cube root of 8, squared: ∛8 = 2, then 2^2 = 4. Taking the root first keeps the numbers small.
| Expression | Root, then power | Value |
|---|---|---|
| 8^(2/3) | ∛8 = 2, 2^2 | 4 |
| 16^(3/4) | ∜16 = 2, 2^3 | 8 |
| (27/8)^(2/3) | ∛27/∛8 = 3/2, (3/2)^2 | 9/4 |
| 2^(3/2) | √2 not rational: √(2^3) = √8 | 2√2 ≈ 2.8284 |
| 4^(-1/2) | 1 / √4 | 1/2 |
| (-8)^(1/3) | cube root of a negative | -2 |
When the answer is exact
The result is a rational number only when the The number being raised to a power, or the side a shape's height is measured from. (numerator and denominator separately) is a perfect nth power. Otherwise the calculator simplifies the A root expression written with the √ symbol, such as a square root. by pulling out perfect powers: 2^(3/2) = √8 = √(4 × 2) = 2√2. The decimal is always shown so you can compare sizes.
Negative bases and negative exponents
A negative base works with an odd root: (-8)^(1/3) = -2. With an even root there is no real answer, because no real number squared is negative. A negative exponent flips the base: 8^(-2/3) = 1 / 8^(2/3) = 1/4. Exponents are reduced first, so (-8)^(2/6) is treated as (-8)^(1/3) = -2.
How to use it
- Type the base as a whole number (8), a fraction (27/8), or a decimal (0.25).
- Enter the exponent's numerator m (negative allowed) and denominator n (the root index).
- Read the exact value when one exists, the radical form otherwise, and the decimal in every case.
- Open Show the work to see the reduced exponent, the root, the power, the The number you get by flipping a fraction; 3/4 becomes 4/3, and 5 becomes 1/5. for negative exponents, and radical simplification.
How to read the answer
'Rational answer: Yes' means the base is a perfect power and the result is an exact fraction. 'No' means the exact answer is a radical, shown in simplest form, and the decimal is an approximation. A negative exponent gives a result less than 1 for bases greater than 1.
Common mistakes and edge cases
- The denominator of the exponent is the root, not a divisor of the base: 8^(1/3) is ∛8 = 2, not 8 ÷ 3.
- A negative exponent does not make the answer negative; it takes a reciprocal.
- An even root of a negative base is not a real number; the calculator says so instead of returning a made-up value.
- Reduce the exponent before deciding whether the root is even: (-8)^(2/6) = (-8)^(1/3) = -2.
- Very large powers overflow exact arithmetic; the calculator then gives the decimal and radical without an exact fraction.
Worked examples
Perfect cube
8^(2/3)
Exact value
4
Fraction base
(27/8)^(2/3)
Exact value
9/4
Radical answer
2^(3/2) = 2√2
Exact value
2·√2
Negative exponent
8^(-2/3)
Exact value
1/4
Negative base, odd root
(-8)^(1/3)
Exact value
-2
Decimal base
0.25^(1/2)
Exact value
1/2
Zero exponent
5^(0/1)
Exact value
1
Not a real number
(-4)^(1/2)
Exact value
Error
Zero to a negative power
0^(-1/2)
Exact value
Error
Frequently asked questions
What does 8^(2/3) equal?+
4. The denominator 3 means cube root and the numerator 2 means square: ∛8 = 2, and 2^2 = 4.
Do I take the root or the power first?+
Either order works mathematically. Taking the root first keeps the numbers small: 16^(3/4) as ∜16 = 2, then 2^3 = 8, instead of 16^3 = 4096, then ∜4096.
How do negative fractional exponents work?+
They mean a reciprocal: x^(-m/n) = 1 / x^(m/n). So 8^(-2/3) = 1/4.
Can the base be negative?+
Yes, when the reduced exponent's denominator is odd: (-8)^(1/3) = -2. With an even root, the result is not a real number.
What if the base is not a perfect power?+
The exact answer is a radical. The calculator simplifies it, for example 2^(3/2) = 2√2, and gives the decimal 2.8284.
How do I enter a fraction as the base?+
Type it as numerator/denominator, such as 27/8. Roots and powers apply to the numerator and denominator separately.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026