8.3 Simplify Rational Exponents

Topics covered in this section are:

Simplify expressions with $a^{\frac{1}{n}}$
Simplify expressions with $a^{\frac{m}{n}}$
Use the properties of exponents to simplify expressions with rational exponents

8.3.1 Simplify Expressions with $\boldsymbol{a^{\frac{1}{n}}}$

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 $(a^{m})^{n}=a^{m \cdot n}$ 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 $(8^{p})^{3}=8$. We will use the Power Property of Exponents to find the value of $p$.

Multiply the exponents on the left.Write the exponent 1 on the right.Since the bases are the same, the exponents must be equal.Solve for $p$.

	$(8^{p})^{3}=8$
Multiply 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 $\left(8^{\frac{1}{3}}\right)^{3}=\sqrt[3]{8}$.

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

RATIONAL EXPONENT $\boldsymbol{a^{\frac{1}{n}}}$

If $\sqrt[n]{a}$ is a real number and $n≥2$, 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 1

Write as a radical expression:

$x^{\frac{1}{2}}$
$y^{\frac{1}{3}}$
$z^{\frac{1}{4}}$

Solution

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

Part 1

	$x^{\frac{1}{2}}$
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 $2$.	$\sqrt{x}$

Part 2

	$y^{\frac{1}{3}}$
The denominator of the rational exponent is $3$, so the index of the radical is $3$.	$\sqrt[3]{y}$

Part 3

	$z^{\frac{1}{3}}$
The denominator of the rational exponent is $4$, so the index of the radical is $4$.	$\sqrt[4]{z}$

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 2

Write with a rational exponent:

$\sqrt{5y}$
$\sqrt[3]{4x}$
$3\sqrt[4]{5z}$

Solution

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

Part 1

	$\sqrt{5y}$
No index is shown, so it is $2$. The denominator of the exponent will be $2$. Put parentheses around the entire expression $5y$.	$(5y)^{\frac{1}{2}}$

Part 2

	$\sqrt[3]{4x}$
The index is $3$, so the denominator of the exponent is $3$. Include parentheses $(4x)$.	$(4x)^{\frac{1}{3}}$

Part 3

	$3\sqrt[4]{5z}$
The index is $4$, so the denominator of the exponent is $4$. Put parentheses only around $(5z)$ because $3$ is not under the radical sign.	$3(5z)^{\frac{1}{4}}$

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

Example 3

Simplify:

$25^{\frac{1}{2}}$
$64^{\frac{1}{3}}$
$256^{\frac{1}{4}}$

Solution

Part 1

	$25^{\frac{1}{2}}$
Rewrite as a square root.	$\sqrt{25}$
Simplify.	$5$

Part 2

	$64^{\frac{1}{3}}$
Rewrite as a cube root.	$\sqrt[3]{64}$
Recognize $64$ is a perfect cube.	$\sqrt[3]{4^{3}}$
Simplify.	$4$

Part 3

	$256^{\frac{1}{4}}$
Rewrite as a fourth root.	$\sqrt[4]{256}$
Recognize $256$ is a perfect fourth power.	$\sqrt[4]{4^{4}}$
Simplify.	$4$

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 4

Simplify:

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

Solution

Part 1

	$(-16)^{\frac{1}{4}}$
Rewrite as a fourth root.	$\sqrt[4]{-16}$
	$\sqrt[4]{(-2)^{4}}$
Simplify.	No real solution.

Part 2

	$-16^{\frac{1}{4}}$
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.	$-2$

Part 3

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

8.3.2 Simplify Expressions with $\boldsymbol{a^{\frac{m}{n}}}$

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

$\boldsymbol{a^{\frac{m}{n}} =(\sqrt[n]{a})^{m} \text{ and } a^{\frac{m}{n}}=(a^{m})^{\frac{1}{n}}=\sqrt[n]{a^{m}}}$

This leads us to the definition.

RATIONAL EXPONENT $\boldsymbol{a^{\frac{m}{n}}}$

For any positive integers $m$ and $n$,

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

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 5

Write with a rational exponent:

$\sqrt{y^{3}}$
$(\sqrt[3]{2x})^{4}$
$\sqrt{(\frac{3a}{4b})^{3}}$

Solution

We want to use $a^{\frac{m}{n}}$ to write each radical in the form $a^{\frac{m}{n}}$.

Part 1

	$\sqrt{y^{3}}$
The numerator of the exponent is the exponent, $\textcolor{red}{3}$.
The denominator of the exponent is the index of the radical, $\textcolor{blue}{2}$.	$y^{\frac{3}{2}}$

Part 2

	$(\sqrt[3]{2x})^{4}$
The numerator of the exponent is the exponent, $\textcolor{red}{4}$.
The denominator of the exponent is the index of the radical, $\textcolor{blue}{3}$.	$(2x)^{\frac{4}{3}}$

Part 3

	$\sqrt{(\frac{3a}{4b})^{3}}$
The numerator of the exponent is the exponent, $\textcolor{red}{3}$.
The denominator of the exponent is the index of the radical, $\textcolor{blue}{2}$.	$(\frac{3a}{4b})^{\frac{3}{2}}$

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

Example 6

Simplify:

$125^{\frac{2}{3}}$
$16^{-\frac{3}{2}}$
$32^{-\frac{2}{5}}$

Solution

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

Part 1

	$125^{\frac{2}{3}}$
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.	$(5)^{2}$
	$25$

Part 2 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 of the radical is the denominator of the exponent, $2$.	$\frac{1}{(\sqrt{16})^{3}}$
Simplify.	$\frac{1}{4^{3}}$
	$\frac{1}{64}$

Part 3

	$32^{-\frac{2}{5}}$
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.	$\frac{1}{(\sqrt[5]{(2^{5}})^{2}}$
Simplify.	$\frac{1}{2^{2}}$
	$\frac{1}{4}$

Example 7

Simplify:

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

Solution

Part 1

	$-25^{\frac{3}{2}}$
Rewrite in radical form.	$-(\sqrt{25})^{3}$
Simplify the radical.	$-(5)^{3}$
Simplify.	$-125$

Part 2

	$-25^{-\frac{3}{2}}$
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.	$-\frac{1}{125}$

Part 3

	$(-25)^{-\frac{3}{2}}$
Rewrite in radical form.	$(\sqrt{-25})^{3}$
There is no real number whose square root is $-25$.	Not a real number.

8.3.3 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

Product Property	$a^{m} \cdot a^{n}= a^{m+n}$
Power Property	$(a^{m})^{n}= a^{m\cdot n}$
Product to a Power	$(ab)^{m}=a^{m}b^{m}$
Quotient Property	$\frac{a^{m}}{a^{n}} = a^{m-n}, a≠0$
Zero Exponent Definition	$a^{0}=1, a≠0$
Quotient to a Power Property	$(\frac{a}{b})^{m}=\frac{a^{m}}{b^{m}}, b≠0$
Negative Exponent Property	$a^{-n}=\frac{1}{a^{n}}, a≠0$

We will apply these properties in the next example.

Example 8

Simplify:

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

Solution

Part 1 The Product Property tell us that when we multiply 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.	$x^{\frac{4}{3}}$

Part 2 The Power Property tell us that when we raise a power to a power, we multiply the exponents.

	$(z^{9})^{\frac{2}{3}}$
To raise a power to a power, we multiply the exponents.	$z^{9 \cdot \frac{2}{3}}$
Simplify.	$z^{6}$

Part 3 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.	$\frac{1}{x^{\frac{4}{3}}}$

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 9

Simplify:

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

Solution

Part 1

	$\left(27u^{\frac{1}{2}}\right)^{\frac{2}{3}}$
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 multiply the exponents.	$\left(3^{2}\right)\left(u^{\frac{1}{2}}\right)^{\frac{2}{3}}$
Simplify.	$9u^{\frac{1}{3}}$

Part 2

	$\left(m^{\frac{2}{3}}n^{\frac{1}{2}}\right)^{\frac{3}{2}}$
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.	$mn^{\frac{3}{4}}$

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

Example 10

Simplify:

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

Solution

Part 1

	$\frac{x^{\frac{3}{4}} \cdot x^{-\frac{1}{4}}}{x^{-\frac{6}{4}}}$
Use the Product Property in the numerator to add the exponents.	$\frac{x^{\frac{2}{4}}}{x^{-\frac{6}{4}}}$
Use the Quotient Property to subtract the exponents.	$x^{\frac{8}{4}}$
Simplify.	$x^{2}$

Part 2 Follow the order of operations to simplify inside the parentheses first.

	$\left(\frac{16x^{\frac{4}{3}}y^{-\frac{5}{6}}}{x^{-\frac{2}{3}}y^{\frac{1}{6}}}\right)^{\frac{1}{2}}$
Use the Quotient Property to subtract the exponents.	$\left(\frac{16x^{\frac{6}{3}}}{y^{\frac{6}{6}}}\right)^{\frac{1}{2}}$
Simplify.	$\left(\frac{16x^{2}}{y}\right)^{\frac{1}{2}}$
Use the Product to a Power Property to multiply the exponents.	$\frac{4x}{y^{\frac{1}{2}}}$

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