Rational Exponents

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So far, we have seen exponential expressions such as below.

$3^{2} = 3 \times 3 = 9 5^{3} = 5 \times 5 \times 5 = 125 {(\frac{4}{9})}^{2} = \frac{4^{2}}{9^{2}} = \frac{4 \times 4}{9 \times 9} = \frac{16}{81}$

Notice that each number in the examples above is raised to an exponent (or power) in the form of a whole number. Now, consider the expressions below.

$3^{\frac{2}{3}}, 5^{\frac{1}{4}}, {(\frac{4}{9})}^{\frac{3}{5}}$

Here, the exponents are in the form of a fraction. These are known as rational exponents. In this article, we shall explore such expressions along with their properties and relationship with radical expressions.

Properties of Exponents

Exponents hold several properties that can help us simplify expressions involving rational exponents. By familiarizing ourselves with these rules, we can solve such expressions quickly without the need for lengthy calculations. The table below describes these properties followed by an example.

Property	Derivation	Example
Product Rule	$a^{m} \cdot a^{n} = a^{m + n}$	$2^{3} \cdot 2^{7} = 2^{3 + 7} = 2^{10}$
Power Rule	${(a^{m})}^{n} = a^{m \cdot n}$	${(2^{3})}^{7} = 2^{3 \cdot 7} = 2^{21}$
Product to Power	${(a b)}^{m} = a^{m} b^{m}$	${(10)}^{3} = {(2 \cdot 5)}^{3} = 2^{3} \cdot 5^{3}$
Quotient Rule	$\frac{a^{m}}{a^{n}} = a^{m - n} (a \neq 0)$	$\frac{2^{3}}{2^{7}} = 2^{3 - 7} = 2^{- 4}$
Zero Exponent Rule	$a^{0} = 1 (a \neq 0)$	$2^{0} = 1$
Quotient to Power Rule	${(\frac{a}{b})}^{m} = \frac{a^{m}}{b^{m}} (b \neq 0)$	${(\frac{2}{5})}^{3} = \frac{2^{3}}{5^{3}}$
Negative Exponent Rule	$a^{- n} = \frac{1}{a^{n}} (a \neq 0)$	$2^{- 3} = \frac{1}{2^{3}}$

Rational Exponents and Radicals

Recall the definition of a radical expression.

A radical expression is an expression that contains a radical symbol √ on any index n, $\sqrt[n]{}$ . This is known as a root function. For example,

$\sqrt{2}, \sqrt[3]{5}, \sqrt{x}, e t c .$

Let's say that we are told to solve the product of two radical expressions. For instance,

$\sqrt{23} \times \sqrt{3}$

How would we go about calculating the product of these radical expressions? This can be somewhat difficult due to the presence of radical symbols. However, there is indeed a solution to this problem. In this article, we shall introduce the concept of rational exponents. Rational exponents can be used to write expressions involving radicals. By writing a radical expression in the form of rational exponents, we can easily simplify them. The definition of a rational exponent is explained below.

Rational exponents are defined as exponents that can be expressed in the form $\frac{p}{q}$ , where q ≠ 0.

The general notation of rational exponents is $x^{\frac{m}{n}}$ . Here, x is called the base (any real number) and $\frac{m}{n}$ is a rational exponent.

Rational exponents can also be written as $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ .

This enables us to conduct operations such as exponents, multiplication, and division. To ease ourselves into this subject, let us begin with the following example. Recall that squaring a number and taking the square root of a number are inverse operations. We can investigate such expressions by assuming that fractional exponents behave as integral exponents.

Integral exponents are exponents expressed in the form of an integer.

1. Coming back to the previous example $\sqrt{23} \times \sqrt{3}$ , we can now do the following

$\sqrt{23} \times \sqrt{3} = 23^{\frac{1}{2}} \times 3^{\frac{1}{2}}$

Applying the product to power rule, we obtain

$23^{\frac{1}{2}} \times 3^{\frac{1}{2}} = {(23 \times 3)}^{\frac{1}{2}} = 69^{\frac{1}{2}}$

Now, coming back to the square root, we obtain

$69^{\frac{1}{2}} = \sqrt{69}$

2. Writing the square of a number as a multiplication

${(a^{\frac{1}{2}})}^{2} = a^{\frac{1}{2}} \cdot a^{\frac{1}{2}}$

Now adding the exponents

$a^{\frac{1}{2}} \cdot a^{\frac{1}{2}} = a^{\frac{1}{2} + \frac{1}{2}}$

Simplifying this, we obtain

$a^{\frac{1}{2} + \frac{1}{2}} = a^{1} = a$

Therefore, the square of $a^{\frac{1}{2}}$ equals to a. Thus, $a^{\frac{1}{2}} = \sqrt{a}$

There are two forms of rational exponents to consider in this topic, namely

$a^{\frac{1}{n}}$ and $a^{\frac{m}{n}}$ .

The following section describes how each of these forms is written in terms of radicals.

Forms of Rational Exponents

There are two forms of rational exponents we must consider here. In each case, we shall exhibit the technique used to simplify each form followed by several worked examples to demonstrate each method.

Case 1

If a is a real number and n ≥ 2, then

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

Write the following in their radical form.

$a^{\frac{1}{3}}$ and ${(4 b)}^{\frac{1}{5}}$

Solutions

1. $a^{\frac{1}{3}} = \sqrt[3]{a}$

2. ${(4 b)}^{\frac{1}{5}} = \sqrt[5]{4 b}$

Express the following in their exponential form.

$\sqrt[7]{x}$ and $\sqrt{2 y}$

Solutions

1. $\sqrt[7]{x} = x^{\frac{1}{7}}$

2. $\sqrt{2 y} = {(2 y)}^{\frac{1}{2}}$

Case 2

For any positive integer m and n,

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

a^{\frac{m}{n}} = \sqrt[n]{a^{m}}

Write the following in their radical form.

a^{\frac{2}{3}}

and

{(7 b)}^{\frac{5}{4}}

Solutions

1. $a^{\frac{2}{3}} = \sqrt[3]{a^{2}}$ , which is the same as $a^{\frac{2}{3}} = {(\sqrt[3]{a})}^{2}$ .

2. ${(7 b)}^{\frac{5}{4}} = {(\sqrt[4]{7 b})}^{5}$

By the Power Rule, we obtain

${(\sqrt[4]{7 b})}^{5} = {(\sqrt[4]{7})}^{5} {(\sqrt[4]{b})}^{5}$

Simplifying this further, our final form becomes

${(\sqrt[4]{7})}^{5} {(\sqrt[4]{b})}^{5} = 7 (\sqrt[4]{7}) (\sqrt[4]{b^{5}})$

Express the following in their exponential form

$\sqrt[5]{x^{8}}$ and ${(\sqrt[8]{2 y})}^{3}$

Solutions

1. $\sqrt[5]{x^{8}} = x^{\frac{8}{5}}$

2. ${(\sqrt[8]{2 y})}^{3} = {(2 y)}^{\frac{3}{8}}$

Evaluating Expressions with Rational Exponents

In this section, we shall look at some worked examples that demonstrate how we can solve expressions involving rational exponents.

Evaluate $27^{- \frac{1}{3}}$

Solution

By the Negative Exponent Rule,

27^{- \frac{1}{3}} = \frac{1}{27^{\frac{1}{3}}}

Now, by the definition of Rational Exponents

$\frac{1}{27^{\frac{1}{3}}} = \frac{1}{\sqrt[3]{27}}$

Simplifying this, we obtain

$\frac{1}{\sqrt[3]{27}} = \frac{1}{\sqrt[3]{3^{3}}} = \frac{1}{3}$

Evaluate $64^{\frac{2}{3}}$

Solution

By the Power Rule,

64^{\frac{2}{3}} = 64^{2 \cdot \frac{1}{3}}

Now, with the definition of Rational Exponents

$64^{2 \cdot \frac{1}{3}} = \sqrt[3]{64^{2}}$

Simplifying this yields

$\sqrt[3]{64^{2}} = \sqrt[3]{{(4^{3})}^{2}} = \sqrt[3]{4^{3} \cdot 4^{3}}$

Further tidying up this expression, we have

$\sqrt[3]{4^{3} \cdot 4^{3}} = 4 \cdot 4 = 16$

Real-World Example

The radius, r, of a sphere with volume, V, is given by the formula

$r = {(\frac{3 V}{4 π})}^{\frac{1}{3}}$ .

What is the radius of a ball if its volume is 24 units³?

Example 1, Aishah Amri - StudySmarter Originals

Given the formula above, the radius of a ball whose volume 24 units³ is given by

$r = {(\frac{3 (24)}{4 π})}^{\frac{1}{3}} \Rightarrow r = {(\frac{72}{4 π})}^{\frac{1}{3}} \Rightarrow r = {(\frac{18}{π})}^{\frac{1}{3}} \Rightarrow r = \sqrt[3]{(\frac{18}{π})} \Rightarrow r = 1.789400458 u n i t s$

Thus, the radius is approximately 1.79 units (correct to two decimal places).

Using Properties of Exponents to Simplify Rational Exponents

Now that we have established the properties of exponents above, let us apply these rules towards simplifying rational exponents. Below are some worked examples showing this.

Simplify the following.

$x^{\frac{1}{5}} \cdot x^{\frac{2}{3}}$

Solution

By the Product Rule

$x^{\frac{1}{5}} \cdot x^{\frac{2}{3}} = x^{\frac{1}{5} + \frac{2}{3}} = x^{\frac{13}{15}}$

Simplify the expression below.

${(x^{4})}^{\frac{3}{7}}$

Solution

By the Power Rule

${(x^{4})}^{\frac{3}{7}} = x^{4 \cdot \frac{3}{7}} = x^{\frac{12}{7}}$

Simplify the following.

\frac{x^{\frac{3}{4}}}{x^{\frac{1}{9}}}

Solution

By the Quotient Rule

$\frac{x^{\frac{3}{4}}}{x^{\frac{1}{9}}} = x^{\frac{3}{4} - \frac{1}{9}} = x^{\frac{23}{36}}$

Simplify the expression below.

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

Solution

By the Product to Power Rule

${(x^{\frac{2}{3}} y^{\frac{1}{4}})}^{\frac{1}{2}} = x^{\frac{2}{3} \cdot \frac{1}{2}} \cdot y^{\frac{1}{4} \cdot \frac{1}{2}} = x^{\frac{1}{3}} \cdot y^{\frac{1}{8}}$

Simplify the following

${(\frac{x^{\frac{1}{2}} x^{\frac{3}{4}}}{x^{- \frac{3}{2}} y^{- \frac{5}{4}}})}^{\frac{1}{3}}$

Solution

By the Product Rule

${(\frac{x^{\frac{1}{2}} x^{\frac{3}{4}}}{x^{- \frac{3}{2}} y^{- \frac{5}{4}}})}^{\frac{1}{3}} = {(\frac{{x^{\frac{1}{2}}}^{+ \frac{3}{4}}}{x^{- \frac{3}{2}} y^{- \frac{5}{4}}})}^{\frac{1}{3}} = {(\frac{x^{\frac{5}{4}}}{x^{- \frac{3}{2}} y^{- \frac{5}{4}}})}^{\frac{1}{3}}$

Followed by the Quotient Rule

${(\frac{x^{\frac{5}{4}}}{x^{- \frac{3}{2}} y^{- \frac{5}{4}}})}^{\frac{1}{3}} = {(\frac{x^{\frac{5}{4} - (- \frac{3}{2})}}{y^{- \frac{5}{4}}})}^{\frac{1}{3}} = {(\frac{x^{\frac{11}{4}}}{y^{- \frac{5}{4}}})}^{\frac{1}{3}}$

Next, by the Product to Power Rule

${(\frac{x^{\frac{11}{4}}}{y^{- \frac{5}{4}}})}^{\frac{1}{3}} = \frac{x^{\frac{11}{4} \cdot \frac{1}{3}}}{y^{- \frac{5}{4} \cdot \frac{1}{3}}} = \frac{x^{\frac{11}{12}}}{y^{- \frac{5}{12}}}$

Finally, by the Negative Exponent Rule

$\frac{x^{\frac{11}{12}}}{y^{- \frac{5}{12}}} = \frac{x^{\frac{11}{12}}}{(\frac{1}{y^{\frac{5}{12}}})} = x^{\frac{11}{12}} \cdot y^{\frac{5}{12}}$

Expressions with Rational Exponents

To determine whether an expression involving rational exponents is fully simplified, the final solution must satisfy the following conditions:

Condition	Example
No negative exponents are present	Instead of writing 3^–², we should simplify this as $\frac{1}{3^{2}}$ by the Negative Exponent Rule
The denominator is not in the form of a fractional exponent	Given that $\frac{3}{4^{\frac{1}{2}}}$ , we should express this as $\frac{3}{\sqrt{4}}$ by the Definition of Rational Exponents
It is not a complex fraction	Rather than writing $\frac{5}{\frac{3}{\sqrt{2}}}$ , we can simplify this as $\frac{5 \sqrt{2}}{3}$ since $\frac{5}{\frac{3}{\sqrt{2}}} = 5 \div \frac{3}{\sqrt{2}} = 5 \times \frac{\sqrt{2}}{3}$
The index of any remaining radical is the least number possible	Say we have a final result of $\sqrt{32}$ . We can further reduce this by noting that $\sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \sqrt{2} = 4 \sqrt{2}$

Properties of Rational Exponents - Key takeaways

A radical expression is a function that contains a square root.
Rational exponents are exponents that can be expressed in the form $\frac{p}{q}$ , where q ≠ 0.
Forms of rational exponents
Form
Representation
$a^{\frac{1}{n}}$
If a is a real number and $n \geq 2$
$\Rightarrow a^{\frac{1}{n}} = \sqrt[n]{a}$
$a^{\frac{m}{n}}$
For any positive integer m and n
$a^{\frac{m}{n}} = {(\sqrt[n]{a})}^{m}$ or $a^{\frac{m}{n}} = \sqrt[n]{a^{m}}$
Properties of exponents
Property
Derivation
Product Rule
$a^{m} \cdot a^{n} = a^{m + n}$
Power Rule
${(a^{m})}^{n} = a^{m \cdot n}$
Product to Power Rule
${(a b)}^{m} = a^{m} b^{m}$
Quotient Rule
$\frac{a^{m}}{a^{n}} = a^{m - n}$
Zero Exponent Rule
$a^{0} = 1$
Quotient to Power Rule
${(\frac{a}{b})}^{m} = \frac{a^{m}}{b^{m}}$
Negative Exponent Rule
$a^{- n} = \frac{1}{a^{n}}$

Frequently Asked Questions about Rational Exponents

Product property, power property, product to a power, quotient property, zero exponent definition, quotient to a power property, negative exponent property

We apply properties of rational exponents to simplify expressions that involve rational exponents

Product rule, power rule, product to a power, quotient property, zero exponent rule, quotient to a power rule, negative exponent rule

Rewrite exponential expressions (the exponent can be a fraction in this case) using the properties of rational exponents

We need rational exponents to solve radical functions

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