Fraction Exponent Calculator

Fraction Exponent Calculator. 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. Also try the use the Simplify Fractions Calculator.

Results

x^(n/d) Result

Negative Root (±)

Fractional Exponent (decimal)

Equivalent Radical Form

What is a fractional exponent?

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. See also our Quotient Calculator.

How is x^(n/d) related to radicals?

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.

What does an exponent of 1/2 mean?

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.

What does a negative fractional exponent mean?

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.

Why does the calculator show a ± result?

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.

Can I use a negative base number?

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.

How do I calculate a regular square root using this calculator?

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.

How do I use this calculator for a regular (non-fractional) exponent?

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. You might also find our Ratio Calculator useful.