Definition of Fractional Exponents

Fractional exponents are a way to represent powers and roots at the same time. When an exponent is fractional, the numerator is the power and the denominator is the root.

For example, x^3⁄₂ = ²√(x³). We can see that the numerator of the fractional exponent is 3 which raises x to the third power. The denominator of the fractional exponent is 2 which takes the square root (also called the second root) of x.

The order of applying the power and root to our number or variable does not matter. In the example, we wrote x^3⁄₂ = ²√(x³). This has us evaluating x³ and then taking the square root of that.

We will get the same solution if we write it as x^3⁄₂ = (²√x)³. In this case, we will be evaluating the square root of x, and then raising that result to the third power.

How to Perform Operations on Fractional Exponents

In this section we will go over how to add, subtract, multiply, and divide fractional exponents. Keep in mind that performing these operations on fractional exponents is the same process as normal exponents, with the extra considerations we must have when operating with fractions.

Adding Fractional Exponents

If terms have the same base a and same fractional exponent ⁿ⁄_m , we can add them. The rule is given as:

Ca^n⁄_m + Da^n⁄_m = (C + D)a^n⁄_m

Here's an example of adding fractional exponents:

2x^2⁄₅ + 7x^2⁄₅ = 9x^2⁄₅

Subtracting Fractional Exponents

Subtracting terms with fractional exponents follows the same rules as adding terms with fractional exponents. The terms must have the same base a and the same fractional exponent ⁿ⁄_m . The rule is given as:

Ca^n⁄_m - Da^n⁄_m = (C - D)a^n⁄_m

Here's an example of subtracting fractional exponents:

2x^2⁄₅ - x^2⁄₅ = x^2⁄₅

Multiplying Fractional Exponents

If terms with fractional exponents have the same base a, then we can multiply them by adding the fractional exponents. The rule is given as:

(a^n⁄_m)(a^p⁄_r) = a^{(n⁄_m + p⁄_r)}

Here's an example of multiplying fractional exponents:

(y^4⁄₅)(y^6⁄₅) = y²

Dividing Fractional Exponents

If terms with fractional exponents have the same base a, then we can divide them by subtracting the fractional exponents. The rule is given as:

^(an⁄_m)⁄_(a^p⁄r) = a^{(n⁄_m - p⁄_r)}

Here's an example of dividing fractional exponents:

^(y3⁄₄)⁄_(y^2⁄4) = y^1⁄₄