2.19: Simplifying Rational Exponents

Simplifying Expressions with \(a^{\frac{1}{n}}\)

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 \(p\).

\(\left(8^{p}\right)^{3}=8\)

Multiple the exponents on the left.

\(8^{3p}=8\)

Write the exponent \(1\) on the right.

\(8^{3p}=8^{1}\)

Since the bases are the same, the exponents must be equal.

\(3p=1\)

Solve for \(p\).

\(p=\frac{1}{3}\)

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

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

Rational Exponent \(a^{\frac{1}{n}}\)

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

\(a^{\frac{1}{n}}=\sqrt[n]{a}\)

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 \(\PageIndex{1}\)

Write as a radical expression:

1. \(x^{\frac{1}{2}}\)
2. \(y^{\frac{1}{3}}\)
3. \(z^{\frac{1}{4}}\)

Solution:

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

a.

\(x^{\frac{1}{2}}\)

The denominator of the rational exponent is \(2\), so the index of the radical is \(2\). We do not show the index when it is \(2\).

\(\sqrt{x}\)

b.

\(y^{\frac{1}{3}}\)

The denominator of the exponent is \(3\), so the index is \(3\).

\(\sqrt[3]{y}\)

c.

\(z^{\frac{1}{4}}\)

The denominator of the exponent is \\(4\), so the index is \(4\).

\(\sqrt[4]{z}\)

You Try \(\PageIndex{1}\)

Write as a radical expression:

1. \(t^{\frac{1}{2}}\)
2. \(m^{\frac{1}{3}}\)
3. \(r^{\frac{1}{4}}\)

Answer

1. \(\sqrt{t}\)
2. \(\sqrt[3]{m}\)
3. \(\sqrt[4]{r}\)

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 \(\PageIndex{2}\)

Write with a rational exponent:

1. \(\sqrt{5y}\)
2. \(\sqrt[3]{4 x}\)
3. \(3 \sqrt[4]{5 z}\)

Solution:

We want to write each radical in the form \(a^{\frac{1}{n}}\)

a.

\(\sqrt{5y}\)

No index is shown, so it is \(2\).

The denominator of the exponent will be \(2\).

Put parentheses around the entire expression \(5y\).

\((5 y)^{\frac{1}{2}}\)

b.

\(\sqrt[3]{4 x}\)

The index is \(3\), so the denominator of the exponent is \(3\). Include parentheses \((4x)\).

\((4 x)^{\frac{1}{3}}\)

c.

\(3 \sqrt[4]{5 z}\)

The index is \(4\), so the denominator of the exponent is \(4\). Put parentheses only around the \(5z\) since 3 is not under the radical sign.

\(3(5 z)^{\frac{1}{4}}\)

You Try \(\PageIndex{2}\)

Write with a rational exponent:

1. \(\sqrt[7]{3 k}\)
2. \(\sqrt[4]{5 j}\)
3. \(8 \sqrt[3]{2 a}\)

Answer

1. \((3 k)^{\frac{1}{7}}\)
2. \((5 j)^{\frac{1}{4}}\)
3. \(8(2 a)^{\frac{1}{3}}\)

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

Example \(\PageIndex{3}\)

Simplify:

1. \(25^{\frac{1}{2}}\)
2. \(64^{\frac{1}{3}}\)
3. \(256^{\frac{1}{4}}\)

Solution:

a.

\(25^{\frac{1}{2}}\)

Rewrite as a square root.

\(\sqrt{25}\)

Simplify.

\(5\)

b.

\(64^{\frac{1}{3}}\)

Rewrite as a cube root.

\(\sqrt[3]{64}\)

Recognize \(64\) is a perfect cube.

\(\sqrt[3]{4^{3}}\)

Simplify.

\(4\)

c.

\(256^{\frac{1}{4}}\)

Rewrite as a fourth root.

\(\sqrt[4]{256}\)

Recognize \(256\) is a perfect fourth power.

\(\sqrt[4]{4^{4}}\)

Simplify.

\(4\)

You Try \(\PageIndex{3}\)

Simplify:

1. \(36^{\frac{1}{2}}\)
2. \(8^{\frac{1}{3}}\)
3. \(16^{\frac{1}{4}}\)

Answer

1. \(6\)
2. \(2\)
3. \(2\)

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

Example \(\PageIndex{4}\)

Simplify:

1. \((-16)^{\frac{1}{4}}\)
2. \(-16^{\frac{1}{4}}\)
3. \((16)^{-\frac{1}{4}}\)

Solution:

a.

\((-16)^{\frac{1}{4}}\)

Rewrite as a fourth root.

\(\sqrt[4]{-16}\)

 

Simplify.

There is no real number raised to the fourth power that gives us -16 so this is not a real number.

b.

\(-16^{\frac{1}{4}}\)

The exponent only applies to the \(16\). Rewrite as a fourth root.

\(-\sqrt[4]{16}\)

Rewrite \(16\) as \(2^{4}\)

\(-\sqrt[4]{2^{4}}\)

Simplify.

\(-2\)

c.

\((16)^{-\frac{1}{4}}\)

Rewrite using the property \(a^{-n}=\frac{1}{a^{n}}\).

\(\frac{1}{(16)^{\frac{1}{4}}}\)

Rewrite as a fourth root.

\(\frac{1}{\sqrt[4]{16}}\)

Rewrite \(16\) as \(2^{4}\).

\(\frac{1}{\sqrt[4]{2^{4}}}\)

Simplify.

\(\frac{1}{2}\)

You Try \(\PageIndex{4}\)

Simplify:

1. \((-64)^{-\frac{1}{2}}\)
2. \(-64^{\frac{1}{2}}\)
3. \((64)^{-\frac{1}{2}}\)

Answer

1. Not a real number
2. \(-8\)
3. \(\frac{1}{8}\)

Simplifying Expressions with \(a^{\frac{m}{n}}\)

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

\(a^{\frac{m}{n}}=\left(a^{\frac{1}{n}}\right)^{m}=(\sqrt[n]{a})^{m} \quad \text { and } \quad a^{\frac{m}{n}}=\left(a^{m}\right)^{^{\frac{1}{n}}}=\sqrt[n]{a^{m}}\)

This leads us to the following definition.

Rational Exponent \(a^{\frac{m}{n}}\)

For any positive integers \(m\) and \(n\),

\(a^{\frac{m}{n}}=(\sqrt[n]{a})^{m} \quad \text { and } \quad a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\)

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 \(\PageIndex{5}\)

Write with a rational exponent:

1. \(\sqrt{y^{3}}\)
2. \((\sqrt[3]{2 x})^{4}\)
3. \(\sqrt{\left(\frac{3 a}{4 b}\right)^{3}}\)

Solution:

We want to use \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\) to write each radical in the form \(a^{\frac{m}{n}}\)

a.

Figure \(\PageIndex{1}\)

b.

Figure \(\PageIndex{2}\)

c.

Figure \(\PageIndex{3}\)

You Try \(\PageIndex{5}\)

Write with a rational exponent:

1. \(\sqrt[5]{a^{2}}\)
2. \((\sqrt[3]{5 a b})^{5}\)
3. \(\sqrt{\left(\frac{7 x y}{z}\right)^{3}}\)

Answer

1. \(a^{\frac{2}{5}}\)
2. \((5 a b)^{\frac{5}{3}}\)
3. \(\left(\frac{7 x y}{z}\right)^{\frac{3}{2}}\)

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

Example \(\PageIndex{6}\)

Simplify:

1. \(125^{\frac{2}{3}}\)
2. \(16^{-\frac{3}{2}}\)
3. \(32^{-\frac{2}{5}}\)

Solution:

We will rewrite the expression as a radical first using the defintion, \(a^{\frac{m}{n}}=(\sqrt[n]{a})^{m}\). 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.

\(125^{\frac{2}{3}}\)

The power of the radical is the numerator of the exponent, \(2\). The index of the radical is the denominator of the exponent, \(3\).

\((\sqrt[3]{125})^{2}\)

Simplify.

\((5)^{2}\)

\(25\)

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

\(16^{-\frac{3}{2}}\)

Rewrite using \(a^{-n}=\frac{1}{a^{n}}\)

\(\frac{1}{16^{\frac{3}{2}}}\)

Change to radical form. The power of the radical is the numerator of the exponent, \(3\). The index is the denominator of the exponent, \(2\).

\(\frac{1}{(\sqrt{16})^{3}}\)

Simplify.

\(\frac{1}{4^{3}}\)

\(\frac{1}{64}\)

c.

\(32^{-\frac{2}{5}}\)

Rewrite using \(a^{-n}=\frac{1}{a^{n}}\)

\(\frac{1}{32^{\frac{2}{5}}}\)

Change to radical form.

\(\frac{1}{(\sqrt[5]{32})^{2}}\)

Rewrite the radicand as a power.

\(\frac{1}{\left(\sqrt[5]{2^{5}}\right)^{2}}\)

Simplify.

\(\frac{1}{2^{2}}\)

\(\frac{1}{4}\)

You Try \(\PageIndex{6}\)

Simplify:

1. \(4^{\frac{3}{2}}\)
2. \(27^{-\frac{2}{3}}\)
3. \(625^{-\frac{3}{4}}\)

Answer

1. \(8\)
2. \(\frac{1}{9}\)
3. \(\frac{1}{125}\)

Example \(\PageIndex{7}\)

Simplify:

1. \(-25^{\frac{3}{2}}\)
2. \(-25^{-\frac{3}{2}}\)
3. \((-25)^{\frac{3}{2}}\)

Solution:

a.

\(-25^{\frac{3}{2}}\)

Rewrite in radical form.

\(-(\sqrt{25})^{3}\)

Simplify the radical.

\(-(5)^{3}\)

Simplify.

\(-125\)

b.

\(-25^{-\frac{3}{2}}\)

Rewrite using \(a^{-n}=\frac{1}{a^{n}}\).

\(-\left(\frac{1}{25^{\frac{3}{2}}}\right)\)

Rewrite in radical form.

\(-\left(\frac{1}{(\sqrt{25})^{3}}\right)\)

Simplify the radical.

\(-\left(\frac{1}{(5)^{3}}\right)\)

Simplify.

\(-\frac{1}{125}\)

c.

\((-25)^{\frac{3}{2}}\)

Rewrite in radical form.

\((\sqrt{-25})^{3}\)

There is no real number whose square root is \(-25\).

Not a real number.

You Try \(\PageIndex{7}\)

Simplify:

1. \(-16^{\frac{3}{2}}\)
2. \(-16^{-\frac{3}{2}}\)
3. \((-16)^{-\frac{3}{2}}\)

Answer

1. \(-64\)
2. \(-\frac{1}{64}\)
3. Not a real number

Using 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 Exponents 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

Product Property

\(a^{m} \cdot a^{n}=a^{m+n}\)

Power Property

\(\left(a^{m}\right)^{n}=a^{m \cdot n}\)

Product to a Power

\((a b)^{m}=a^{m} b^{m}\)

Quotient Property

\(\frac{a^{m}}{a^{n}}=a^{m-n}, a \neq 0\)

Zero Exponent Definition

\(a^{0}=1, a \neq 0\)

Quotient to a Power Property

\(\left(\frac{a}{b}\right)^{m}=\frac{a^{m}}{b^{m}}, b \neq 0\)

Negative Exponent Property

\(a^{-n}=\frac{1}{a^{n}}, a \neq 0\)

We will apply these properties in the next example.

Example \(\PageIndex{8}\)

Simplify:

1. \(x^{\frac{1}{2}} \cdot x^{\frac{5}{6}}\)
2. \(\left(z^{9}\right)^{\frac{2}{3}}\)
3. \(\frac{x^{\frac{1}{3}}}{x^{\frac{5}{3}}}\)

Solution

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

\(x^{\frac{1}{2}} \cdot x^{\frac{5}{6}}\)

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

\(x^{\frac{1}{2}+\frac{5}{6}}\)

Add the fractions.

\(x^{\frac{8}{6}}\)

Simplify the exponent.

\(x^{\frac{4}{3}}\)

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

\(\left(z^{9}\right)^{\frac{2}{3}}\)

To raise a power to a power, we multiple the exponents.

\(z^{9 \cdot \frac{2}{3}}\)

Simplify.

\(z^{6}\)

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

\(\frac{x^{\frac{1}{3}}}{x^{\frac{5}{3}}}\)

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

\(\frac{1}{x^{\frac{5}{3}-\frac{1}{3}}}\)

Simplify.

\(\frac{1}{x^{\frac{4}{3}}}\)

You Try \(\PageIndex{8}\)

Simplify:

1. \(y^{\frac{3}{4}} \cdot y^{\frac{5}{8}}\)
2. \(\left(m^{9}\right)^{\frac{2}{9}}\)
3. \(\frac{d^{\frac{1}{5}}}{d^{\frac{6}{5}}}\)

Answer

1. \(y^{\frac{11}{8}}\)
2. \(m^{2}\)
3. \(\frac{1}{d}\)

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 \(\PageIndex{9}\)

Simplify:

1. \(\left(27 u^{\frac{1}{2}}\right)^{\frac{2}{3}}\)
2. \(\left(m^{\frac{2}{3}} n^{\frac{1}{2}}\right)^{\frac{3}{2}}\)

Solution:

a.

\(\left(27 u^{\frac{1}{2}}\right)^{\frac{2}{3}}\)

First we use the Product to a Power Property.

\((27)^{\frac{2}{3}}\left(u^{\frac{1}{2}}\right)^{\frac{2}{3}}\)

Rewrite \(27\) as a power of \(3\).

\(\left(3^{3}\right)^{\frac{2}{3}}\left(u^{\frac{1}{2}}\right)^{\frac{2}{3}}\)

To raise a power to a power, we multiple the exponents.

\(\left(3^{2}\right)\left(u^{\frac{1}{3}}\right)\)

Simplify.

\(9 u^{\frac{1}{3}}\)

b.

\(\left(m^{\frac{2}{3}} n^{\frac{1}{2}}\right)^{\frac{3}{2}}\)

First we use the Product to a Power Property.

\(\left(m^{\frac{2}{3}}\right)^{\frac{3}{2}}\left(n^{\frac{1}{2}}\right)^{\frac{3}{2}}\)

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

\(m n^{\frac{3}{4}}\)

You Try \(\PageIndex{9}\)

Simplify:

1. \(\left(81 n^{\frac{2}{5}}\right)^{\frac{3}{2}}\)
2. \(\left(a^{\frac{3}{2}} b^{\frac{1}{2}}\right)^{\frac{4}{3}}\)

Answer

1. \(729 n^{\frac{3}{5}}\)
2. \(a^{2} b^{\frac{2}{3}}\)

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

Example \(\PageIndex{10}\)

Simplify:

1. \(\frac{x^{\frac{3}{4}} \cdot x^{-\frac{1}{4}}}{x^{-\frac{6}{4}}}\)
2. \(\left(\frac{16 x^{\frac{4}{3}} y^{-\frac{5}{6}}}{x^{-\frac{2}{3}} y^{\frac{1}{6}}}\right)^{\frac{1}{2}}\)

Solution:

a.

\(\frac{x^{\frac{3}{4}} \cdot x^{-\frac{1}{4}}}{x^{-\frac{6}{4}}}\)

Use the Product Property in the numerator, add the exponents.

\(\frac{x^{\frac{2}{4}}}{x^{-\frac{6}{4}}}\)

Use the Quotient Property, subtract the exponents.

\(x^{\frac{8}{4}}\)

Simplify.

\(x^{2}\)

b.

\(\left(\frac{16 x^{\frac{4}{3}} y^{-\frac{5}{6}}}{x^{-\frac{2}{3}} y^{\frac{1}{6}}}\right)^{\frac{1}{2}}\)

Use the Quotient Property, subtract the exponents.

\(\left(\frac{16 x^{\frac{6}{3}}}{y^{\frac{6}{6}}}\right)^{\frac{1}{2}}\)

Simplify.

\(\left(\frac{16 x^{2}}{y}\right)^{\frac{1}{2}}\)

Use the Product to a Power Property, multiply the exponents.

\(\frac{4 x}{y^{\frac{1}{2}}}\)

You Try \(\PageIndex{10}\)

Simplify:

1. \(\frac{m^{\frac{2}{3}} \cdot m^{-\frac{1}{3}}}{m^{-\frac{5}{3}}}\)
2. \(\left(\frac{25 m^{\frac{1}{6}} n^{\frac{11}{6}}}{m^{\frac{2}{3}} n^{-\frac{1}{6}}}\right)^{\frac{1}{2}}\)

Answer

1. \(m^{2}\)
2. \(\frac{5 n}{m^{\frac{1}{4}}}\)