Simplify Rational Exponents

Simplify Expressions with $a^{\tfrac{1}{n}}$

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.

$\begin{aligned} & & & & & \left(8^{p}\right)^{3} & = & 8\ \\ \text{Multiply the exponents on the left}. & & & & & 8^{3 p} & = & 8\ \\ \text{Write the exponent 1 on the right}. & & & & & 8^{3 p} & = & 8^{1}\ \\ \text{Since the bases are the same},\text{ the exponents must be equal}. & & & & & 3 p & = & 1\ \\ \text{Solve for}\ p. & & & & & p & = & \tfrac{1}{3}\end{aligned}$

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

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

Rational Exponent $a^{\tfrac{1}{n}}$

If $\sqrt[n]{a}$ is a real number and $n \geq 2,$ then

The denominator of the rational exponent is the index of the radical.

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) $x^{\tfrac{1}{2}}$ (b) $y^{\tfrac{1}{3}}$ (c) $z^{\tfrac{1}{4}}.$

We want to write each expression in the form $\sqrt[n]{a}.$

(a)

(b)

(c)

In the next example, we will write each radical using a rational exponent. It is important to use parentheses around the entire expression in the radicand since the entire expression is raised to the rational power.

Example.

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

We want to write each radical in the form $a^{\tfrac{1}{n}}.$

(a)

(b)

(c)

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

Example.

Simplify: (a) $25^{\tfrac{1}{2}}$ (b) $64^{\tfrac{1}{3}}$ (c) $256^{\tfrac{1}{4}}.$

(a)

(b)

(c)

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) $\left(-16\right)^{\tfrac{1}{4}}$ (b) $- 16^{\tfrac{1}{4}}$ (c) $\left(16\right)^{- \tfrac{1}{4}}.$

(a)

(b)

(c)

Simplify Expressions with $a^{\tfrac{m}{n}}$

We can look at $a^{\tfrac{m}{n}}$ in two ways. Remember the Power Property tells us to multiply the exponents and so $\left(a^{\tfrac{1}{n}}\right)^{m}$ and $\left(a^{m}\right)^{^{\tfrac{1}{n}}}$ both equal $a^{\tfrac{m}{n}}.$ If we write these expressions in radical form, we get

This leads us to the following definition.

Rational Exponent $a^{\tfrac{m}{n}}$

For any positive integers m and n,

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, before raising it to the power indicated.

Example.

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

We want to use $a^{\tfrac{m}{n}} = \sqrt[n]{a^{m}}$ to write each radical in the form $a^{\tfrac{m}{n}}.$

(a)

The exponent 3 on the radicand becomes the numerator of the rational exponent, and the square root’s index 2 becomes the denominator:

(b)

The outside exponent 4 becomes the numerator, and the cube root’s index 3 becomes the denominator:

(c)

The power 3 becomes the numerator, and the square root’s index 2 becomes the denominator:

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

Example.

Simplify: (a) $125^{\tfrac{2}{3}}$ (b) $16^{- \tfrac{3}{2}}$ (c) $32^{- \tfrac{2}{5}}.$

We will rewrite the expression as a radical first using the defintion, $a^{\tfrac{m}{n}} = \left(\sqrt[n]{a}\right)^{m}.$ This form lets us take the root first and so we keep the numbers in the radicand smaller than if we used the other form.

(a)

(b) We will rewrite each expression first using $a^{- n} = \tfrac{1}{a^{n}}$ and then change to radical form.

(c)

Example.

Simplify: (a) $- 25^{\tfrac{3}{2}}$ (b) $- 25^{- \tfrac{3}{2}}$ (c) $\left(-25\right)^{\tfrac{3}{2}}.$

(a)

(b)

(c)

Use the Properties of Exponents to Simplify Expressions with Rational Exponents

The same properties of exponents that we have already used also apply to rational exponents. We will list the Properties of Exponenets here to have them for reference as we simplify expressions.

Properties of Exponents

If a and b are real numbers and m and n are rational numbers, then

$\begin{aligned}\textbf{Product Property} & & & a^{m} \cdot a^{n} & = & a^{m + n}\ \\ \textbf{Power Property} & & & \left(a^{m}\right)^{n} & = & a^{m \cdot n}\ \\ \textbf{Product to a Power} & & & \left(a b\right)^{m} & = & a^{m} b^{m}\ \\ \textbf{Quotient Property} & & & \tfrac{a^{m}}{a^{n}} & = & a^{m - n}, a \neq 0\ \\ \textbf{Zero Exponent Definition} & & & a^{0} & = & 1, a \neq 0\ \\ \textbf{Quotient to a Power Property} & & & \left(\tfrac{a}{b}\right)^{m} & = & \tfrac{a^{m}}{b^{m}}, b \neq 0\ \\ \textbf{Negative Exponent Property} & & & a^{- n} & = & \tfrac{1}{a^{n}}, a \neq 0\end{aligned}$

We will apply these properties in the next example.

Example.

Simplify: (a) $x^{\tfrac{1}{2}} \cdot x^{\tfrac{5}{6}}$ (b) $\left(z^{9}\right)^{\tfrac{2}{3}}$ (c) $\tfrac{x^{\tfrac{1}{3}}}{x^{\tfrac{5}{3}}}.$

(a) The Product Property tells us that when we multiply the same base, we add the exponents.

(b) The Power Property tells us that when we raise a power to a power, we multiply the exponents.

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

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

Example.

Simplify: (a) $\left(27 u^{\tfrac{1}{2}}\right)^{\tfrac{2}{3}}$ (b) $\left(m^{\tfrac{2}{3}} n^{\tfrac{1}{2}}\right)^{\tfrac{3}{2}}.$

(a)

(b)

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

Example.

Simplify: (a) $\tfrac{x^{\tfrac{3}{4}} \cdot x^{- \tfrac{1}{4}}}{x^{- \tfrac{6}{4}}}$ (b) $\left(\tfrac{16 x^{\tfrac{4}{3}} y^{- \tfrac{5}{6}}}{x^{- \tfrac{2}{3}} y^{\tfrac{1}{6}}}\right)^{\tfrac{1}{2}}.$

(a)

(b) Follow the order of operations to simplify inside the parenthese first.

This section is adapted from Intermediate Algebra 2e, Section 8.3: Simplify Rational Exponents by Lynn Marecek and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: reformatted worked solutions for the web; omitted the Be Prepared quiz, media links, self-check reflection, and end-of-section exercise bank; and converted the source Try It practice into interactive exercises.