Rational Exponents Calculator

Background.

A fractional exponent packs two operations into one symbol. In a^(m/n), the denominator n says which root to take and the numerator m says what power to raise it to, so 27^(2/3) means the cube root of 27, squared: 3 squared, or 9. The notation is not a textbook convenience; it is what keeps the ordinary exponent rules working. Since a^(1/2) × a^(1/2) must equal a¹, a^(1/2) has to be the square root of a.

Rational exponents turn up wherever a quantity scales by a power that is not a whole number. Kepler's third law says a planet's orbital period in years equals its distance from the Sun in astronomical units raised to the 3/2 power, so a planet 4 AU out takes 4^(3/2) = 8 years to go round. Something that grew by half over five years grew at an average of 1.5^(1/5) − 1, about 8.4% a year. In music, each semitone of an equal-tempered scale multiplies the pitch frequency by 2^(1/12), about 1.0595.

The base here must be positive. With a negative base the answer can depend on how the fraction is written: (−8)^(1/3) is −2, yet reading (−8)^(2/6) as the sixth root of (−8)², which is 64, gives +2, even though 1/3 and 2/6 are equal. Keeping the base positive guarantees a single real answer, the principal root, and means an unreduced fraction such as 4/6 gives exactly the same result as 2/3.

A negative numerator turns the result into a reciprocal: 8^(−2/3) is 1 ÷ 8^(2/3), or 0.25. Both parts of the exponent must be whole numbers, so an exponent of 0.75 is entered as 3 over 4.

What is rational exponents calculator?

A rational exponent is an exponent written as a fraction m/n of two integers. For a positive base a, a^(m/n) is defined as the n-th root of a raised to the m-th power, which is the same as the n-th root of a^m. The result is the principal, positive real value. Whole-number exponents are the special case n = 1, and a numerator of zero gives 1 whatever the base.

How to use this calculator.

1. 01Enter the Positive base, the number under the root. It must be greater than zero, since a negative base can give different answers for fractions that are equal.
2. 02Enter the Exponent numerator, the power to raise to. A negative whole number gives a reciprocal: −2 over 3 returns 1 ÷ a^(2/3).
3. 03Enter the Exponent denominator, the root to take — 2 for a square root, 3 for a cube root — as a whole number from 1 to 1000. Convert a decimal exponent to a fraction first: 0.75 is 3/4 and 1.5 is 3/2.
4. 04Read the value, shown to at most 10 decimal places. A result smaller than about 5 × 10⁻¹¹ therefore shows as 0, and one larger than about 1.8 × 10³⁰⁸ is refused as too large to display.

The formula.

a^(m/n) = (ⁿ√a)^m = ⁿ√(a^m), where a is the positive base, m the integer numerator and n the positive integer denominator. The page divides m by n and raises a to that power directly, which for a positive base equals taking the root and then the power, or the power and then the root. By hand, taking the root first keeps the numbers small: 27^(2/3) is 3² rather than ∛729. The denominator pulls the result towards 1 — the larger n is, the closer ⁿ√a sits to 1, so 2^(1/12) is only about 1.0595 — while the numerator multiplies the effect. A negative m gives the reciprocal. When the true result is too large to represent, as 27^(1000/3) = 3^1000, about 1.32 × 10⁴⁷⁷, is, the page refuses it rather than showing an infinite or wrong figure.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Rice University. College Algebra 2e, 2021. Algebra and function chapters. Retrieved 2026-08-06.
2. [2]OpenStax, Rice University. Precalculus 2e, 2021. Trigonometry and analytic geometry chapters. Retrieved 2026-08-06.
3. [3]NIST/SEMATECH. e-Handbook of Statistical Methods, 2012. Chapter 1.3 and Chapter 4. Retrieved 2026-08-06.