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Rational Exponents

Rational Exponents

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

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.

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
Power Rule
Product to Power
Quotient Rule
Zero Exponent Rule
Quotient to Power Rule
Negative Exponent Rule

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, . This is known as a root function. For example,

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

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 , where q ≠ 0.

The general notation of rational exponents is . Here, x is called the base (any real number) and is a rational exponent.

Rational exponents can also be written as .

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 , we can now do the following

Applying the product to power rule, we obtain

Now, coming back to the square root, we obtain

2. Writing the square of a number as a multiplication

Now adding the exponents

Simplifying this, we obtain

Therefore, the square of equals to a. Thus,

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

and .

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

.

Write the following in their radical form.

and

Solutions

1.

2.

Express the following in their exponential form.

and

Solutions

1.

2.

Case 2

For any positive integer m and n,

or ,

Write the following in their radical form.

and

Solutions

1. , which is the same as .

2.

By the Power Rule, we obtain

Simplifying this further, our final form becomes

Express the following in their exponential form

and

Solutions

1.

2.

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

Solution

By the Negative Exponent Rule,

Now, by the definition of Rational Exponents

Simplifying this, we obtain

Evaluate

Solution

By the Power Rule,

Now, with the definition of Rational Exponents

Simplifying this yields

Further tidying up this expression, we have

Real-World Example

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

.

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

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.

Solution

By the Product Rule

Simplify the expression below.

Solution

By the Power Rule

Simplify the following.

Solution

By the Quotient Rule

Simplify the expression below.

Solution

By the Product to Power Rule

Simplify the following

Solution

By the Product Rule

Followed by the Quotient Rule

Next, by the Product to Power Rule

Finally, by the Negative Exponent Rule

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 32, we should simplify this as by the Negative Exponent Rule

The denominator is not in the form of a fractional exponent

Given that , we should express this as by the Definition of Rational Exponents

It is not a complex fraction

Rather than writing , we can simplify this as since

The index of any remaining radical is the least number possible

Say we have a final result of . We can further reduce this by noting that

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 , where q ≠ 0.
  • Forms of rational exponents
    Form
    Representation
    If a is a real number and
    For any positive integer m and n
    or
  • Properties of exponents
    Property
    Derivation
    Product Rule
    Power Rule
    Product to Power Rule
    Quotient Rule
    Zero Exponent Rule
    Quotient to Power Rule
    Negative Exponent Rule

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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