Fraction Exponent Calculator (x^(a/b) as a root, with both methods)

Understanding the fraction exponent calculator — concept and principle

This calculator evaluates a number raised to a fractional exponent, x^(a/b), and shows both routes: root first, (ᵇ√x)ᵃ, and power first, ᵇ√(xᵃ). It reduces the exponent, gives the exact simplified form and the radical form. Use it for algebra homework on rational exponents and radicals.

The denominator is the root and the numerator is the power; order does not matter.

How it is calculated

x^(a/b) = (ᵇ√x)ᵃ = ᵇ√(xᵃ)

The denominator b is the index of the root; the numerator a is the power.
The exponent is reduced to lowest terms first.
A negative exponent means the reciprocal: x^(−a/b) = 1 ÷ x^(a/b).
For a negative base, a real value exists only when b is odd.
For whole-number bases, perfect powers are taken out of the root, such as 32^(3/2) = 128√2.

The principle

The rule (x^m)^n = x^(mn) extends naturally to fractions. Since (x^(1/b))^b = x, the number x^(1/b) must be the b-th root of x. Then x^(a/b) = (x^(1/b))^a, the root raised to the power a. Taking the root first keeps the numbers small, which is useful by hand; the power-first route gives the same answer.

Worked example

Default inputs: x = 8, exponent 2/3.

Exponent in lowest terms: 2/3.
Root first: (∛8)² = 2² = 4.
Power first: ∛(8²) = ∛64 = 4.
Value of 8^(2/3) = 4; exact form 4.
Written as a radical: ∛(8²).

Each point is the calculator re-run with only “Base x” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Base x	Value of x^(a/b)
2	1.587401052
4	2.5198421
6	3.301927249
8	4
10	4.641588834
12	5.241482788
16	6.349604208

How to read the result

A fractional exponent between 0 and 1 gives a value smaller than x for x > 1; an exponent above 1 gives a larger value. 8^(2/3) = 4 sits between 8^0 = 1 and 8^1 = 8, as expected. With a negative exponent, such as 8^(−2/3), the result is the reciprocal, 1/4.

Common mistakes

Swapping numerator and denominator: 8^(2/3) is not √(8³).
Treating a negative exponent as a negative result.
Taking an even root of a negative base, which has no real value.
Forgetting to reduce the exponent, such as 4/6 to 2/3, before deciding whether a negative base is allowed.

Keep in mind: Calculators and software may treat negative bases with fractional exponents differently; this page follows the real-root convention with the reduced exponent.

Frequently asked questions

What does a fractional exponent mean?

x^(a/b) is the b-th root of x raised to the power a.

What is 8^(2/3)?

4, because the cube root of 8 is 2 and 2² = 4.

What is 27^(−1/3)?

1/3, the reciprocal of the cube root of 27.

Sources: OpenStax, Intermediate Algebra 2e, §8.3 Simplify Rational Exponents

Checked 9 Nov 2026. OpenStax, Intermediate Algebra 2e (2020), §8.3 Simplify Rational Exponents (a^(1/n) = ⁿ√a; a^(m/n) = (ⁿ√a)^m = ⁿ√(a^m) for positive integers m and n, taking the root first keeps numbers small; even roots of negative numbers are not real, e.g. (−16)^(1/4)).

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