Enter a base number (x), a numerator (n), and a denominator (d) to compute x raised to the fractional power n/d — equivalent to taking the d-th root of x raised to n. The Fraction Exponent Calculator returns the result, the equivalent radical form, and both positive and negative roots where applicable.
x^(n/d) Result
--
Negative Root (±)
--
Fractional Exponent (decimal)
--
Equivalent Radical Form
--
A fractional exponent expresses both a power and a root in one notation. For a fraction n/d, x^(n/d) means you raise x to the power n and then take the d-th root — or equivalently take the d-th root first and then raise to the power n. For example, 4^(1/2) is the square root of 4, which equals 2.
x^(n/d) is equivalent to the d-th root of x raised to the n-th power: x^(n/d) = ᵈ√(xⁿ). You can also write it as (ᵈ√x)ⁿ — both forms give the same result. For example, 4^(3/2) = √(4³) = √64 = 8.
An exponent of 1/2 is simply the square root. So x^(1/2) = √x. Similarly, 1/3 means cube root, 1/4 means fourth root, and so on — an exponent of 1/k is the k-th root of x.
A negative fractional exponent means you take the reciprocal first. So x^(-n/d) = 1 / x^(n/d). For example, 4^(-1/2) = 1/√4 = 0.5. This calculator handles negative numerators automatically.
When the denominator d is even and the numerator n is odd, the d-th root of a positive number has both a positive and negative real solution. For instance, 4^(3/2) = ±8 because both (+8)² = 64 and (-8)² = 64. The calculator returns both the positive primary result and the corresponding negative root.
Negative bases can produce real results when the root index (denominator) is odd. For example, (-8)^(1/3) = -2. However, even roots of negative numbers produce complex (imaginary) results, which this calculator does not support — it will show NaN in those cases.
Set the numerator (n) to 1 and the denominator (d) to 2. Enter your base number and the result will be its square root. For the cube root, use n=1 and d=3. For any k-th root, use n=1 and d=k.
Set the denominator (d) to 1 and enter any integer as the numerator (n). Then x^(n/1) = xⁿ, a standard exponent. For example, x=2, n=8, d=1 calculates 2⁸ = 256.