Fraction Exponent Calculator — Evaluate x^(n/d), Roots, Negative & Fractional Powers Step by Step

Fraction Exponent Calculator – How to Use

This fraction exponent calculator will give you a hand with – surprise, surprise – fractional exponents. Do you struggle with the concept of fractional exponents? Are negative and fractional exponents a closed book to you? Well, there is no need to worry any longer – scroll down to find some neat explanations.

Just enter the base and the numerator and denominator of the exponent, and the tool instantly returns the value of xn/d, together with the equivalent radical form and a full step-by-step breakdown.

Fractional exponents with the numerator equal to 1

Fractional exponents are a way of expressing powers as well as roots in one notation. Let's have a look at a few simple examples first, where the numerator is equal to 1:

• 641/2 = √64 = 8
• 271/3 = ³√27 = 3

From the equations above we can deduce that:

• An exponent of 1/2 is a square root
• An exponent of 1/3 is a cube root
• An exponent of 1/4 is a fourth root
• …and, in general, an exponent of 1/k is a k-th root:

x1/k = k√x

Why is that so? Using the law of exponents that lets us add exponents when multiplying powers with the same base, x = x1/2 + 1/2 = x1/2 × x1/2. A number which, when multiplied by itself, gives x is precisely the square root of x – so x1/2 = √x.

Fractional exponents with a numerator different from 1 (any fraction)

So what happens if the numerator is not equal to 1? For a = xn/d all you need to do is raise the number to the power n and take the d-th root. The order doesn't matter, because the fraction n/d may be split into a whole number n and a fraction part 1/d:

xn/d = (xn)1/d = d√(xn) = (d√x)n

Let's look at an example where the fractional exponent is 3/2 and x = 16:

• 163/2 = (163)1/2 = √(4096) = 64
• 163/2 = (161/2)3 = (√16)3 = 43 = 64

The result is the same – choose whichever method gives you the easiest computation.

Negative and fractional exponents

Positive exponents tell us how many times we use a number in a multiplication. A negative exponent means the reciprocal: x-n/d = 1 ÷ xn/d. Combining the two rules, a negative fractional exponent is simply the reciprocal of the corresponding root and power. For example:

• 8-2/3 = 1 ÷ 82/3 = 1 ÷ (³√8)2 = 1 ÷ 4 = 0.25

Good to know

• A negative base gives a real result only when the reduced denominator is odd (e.g. (-8)1/3 = -2). An even root of a negative number is not a real number.
• 0 raised to a positive exponent is 0; 00 and 0 to a negative exponent are undefined.
• The calculator automatically reduces the exponent fraction to lowest terms before evaluating.

If you first need to review some basics about exponentiation, visit our exponent calculator, and for a refresher on roots, take a look at our root, square root and cube root calculators.