Rational Exponents

Simplify expressions with $a^{1/n}$

Rational exponents are another way of writing expressions with radicals. When we use rational exponents, we can apply the properties of exponents to simplify expressions.

The Power Property for Exponents says that $\left(a^m\right)^n = a^{m \cdot n}$ when $m$ and $n$ are whole numbers. Let’s assume we are now not limited to whole numbers.

Suppose we want to find a number $p$ such that $\left(8^p\right)^3 = 8$. We will use the Power Property of Exponents to find the value of $p$:

So $\left(8^{1/3}\right)^3 = 8$. But we know also that $\left(\sqrt[3]{8}\right)^3 = 8$. Then it must be that $8^{1/3} = \sqrt[3]{8}$.

This same logic can be used for any positive integer exponent $n$ to show that $a^{1/n} = \sqrt[n]{a}$.

There will be times when working with expressions will be easier if you use rational exponents, and times when it will be easier if you use radicals. In the first few examples, you’ll practice converting expressions between these two notations.

Example. Write as a radical expression: (a) $x^{1/2}$, (b) $y^{1/3}$, (c) $z^{1/4}$.

We want to write each expression in the form $\sqrt[n]{a}$. The denominator of the rational exponent is the index of the radical:

We do not show the index when it is $2$.

Example. Write with a rational exponent: (a) $\sqrt{x}$, (b) $\sqrt[3]{y}$, (c) $\sqrt[4]{z}$.

We want to write each radical in the form $a^{1/n}$. The index of the radical becomes the denominator of the rational exponent:

Example. Write with a rational exponent: (a) $\sqrt{5y}$, (b) $\sqrt[3]{4x}$, (c) $3\sqrt[4]{5z}$.

The entire radicand becomes the base. Note in part (c) that the factor $3$ is not under the radical, so it stays outside:

In the next example, you may find it easier to simplify the expressions if you rewrite them as radicals first.

Example. Simplify: (a) $25^{1/2}$, (b) $64^{1/3}$, (c) $256^{1/4}$.

Rewrite each as a radical, then recognize the perfect power and simplify:

Be careful of the placement of the negative signs in the next example. We will need to use the property $a^{-n} = \tfrac{1}{a^n}$ in one case.

Example. Simplify: (a) $(-64)^{1/3}$, (b) $-64^{1/3}$, (c) $(64)^{-1/3}$.

Example. Simplify: (a) $(-16)^{1/4}$, (b) $-16^{1/4}$, (c) $(16)^{-1/4}$.

Simplify expressions with $a^{m/n}$

Let’s work with the Power Property for Exponents some more. Suppose we raise $a^{1/n}$ to the power $m$:

So $a^{m/n} = \left(\sqrt[n]{a}\right)^m$. Now suppose we take $a^m$ to the $\tfrac{1}{n}$ power:

So $a^{m/n} = \sqrt[n]{a^m}$ also. Which form do we use to simplify an expression? We usually take the root first — that way we keep the numbers in the radicand smaller.

Example. Write with a rational exponent: (a) $\sqrt{y^3}$, (b) $\sqrt[3]{x^2}$, (c) $\sqrt[4]{z^3}$.

We use $a^{m/n} = \sqrt[n]{a^m}$: the exponent in the radicand is the numerator, and the index is the denominator:

Example. Simplify: (a) $9^{3/2}$, (b) $125^{2/3}$, (c) $81^{3/4}$.

We rewrite each expression as a radical first using $a^{m/n} = \left(\sqrt[n]{a}\right)^m$. This form lets us take the root first, keeping the numbers in the radicand smaller:

Remember that $b^{-p} = \tfrac{1}{b^p}$. The negative sign in the exponent does not change the sign of the expression.

Example. Simplify: (a) $16^{-3/2}$, (b) $32^{-2/5}$, (c) $4^{-5/2}$.

We rewrite each expression first using $b^{-p} = \tfrac{1}{b^p}$ and then change to radical form:

Example. Simplify: (a) $-25^{3/2}$, (b) $-25^{-3/2}$, (c) $(-25)^{3/2}$.

Use the laws of exponents to simplify expressions with rational exponents

The same laws of exponents that we already used apply to rational exponents, too. We list the exponent properties here for reference as we simplify expressions.

When we multiply the same base, we add the exponents.

Example. Simplify: (a) $2^{1/2} \cdot 2^{5/2}$, (b) $x^{2/3} \cdot x^{4/3}$, (c) $z^{3/4} \cdot z^{5/4}$.

The bases are the same, so we add the exponents:

We will use the Power Property in the next example.

Example. Simplify: (a) $\left(x^4\right)^{1/2}$, (b) $\left(y^6\right)^{1/3}$, (c) $\left(z^9\right)^{2/3}$.

To raise a power to a power, we multiply the exponents:

The Quotient Property tells us that when we divide with the same base, we subtract the exponents.

Example. Simplify: (a) $\tfrac{x^{4/3}}{x^{1/3}}$, (b) $\tfrac{y^{3/4}}{y^{1/4}}$, (c) $\tfrac{z^{2/3}}{z^{5/3}}$.

To divide with the same base, we subtract the exponents:

Sometimes we need to use more than one property. In the next examples, we will use both the Product to a Power Property and then the Power Property.

Example. Simplify: (a) $\left(27u^{1/2}\right)^{2/3}$, (b) $\left(8v^{1/4}\right)^{2/3}$.

First we use the Product to a Power Property, then the Power Property:

Example. Simplify: (a) $\left(m^3 n^9\right)^{1/3}$, (b) $\left(p^4 q^8\right)^{1/4}$.

First we use the Product to a Power Property, then the Power Property:

We will use both the Product and Quotient Properties in the next example.

Example. Simplify: (a) $\tfrac{x^{3/4} \cdot x^{-1/4}}{x^{-6/4}}$, (b) $\tfrac{y^{4/3} \cdot y}{y^{-2/3}}$.

Use the Product Property in the numerator (add the exponents), then the Quotient Property (subtract the exponents):

Key terms

rational exponent — an exponent that is a fraction; $a^{1/n} = \sqrt[n]{a}$ and $a^{m/n} = \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}$, connecting exponent notation to radical notation. negative rational exponent — a negative fractional exponent, rewritten with a positive exponent using $a^{-m/n} = \tfrac{1}{a^{m/n}}$.

This section is adapted from Elementary Algebra 2e, 9.8 Rational Exponents by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recast the worked-example step tables as typeset math, condensed the multi-part “Try It” problems into single-part interactive exercises with instant feedback, and omitted the Be Prepared quiz, media links, Self Check, and end-of-section exercises.