Rational Exponents

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

Rational Exponents Rational Exponents

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    32=3×3=953=5×5×5=125492=4292=4×49×9=1681

    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.

    323, 514, 4935

    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.

    PropertyDerivationExample
    Product Ruleam·an=am+n23·27=23+7=210
    Power Rule(am)n=am·n237=23·7=221
    Product to Powerabm=ambm103=2·53=23·53
    Quotient Ruleaman=am-n (a0)2327=23-7=2-4
    Zero Exponent Rulea0=1 (a0)20=1
    Quotient to Power Ruleabm=ambm (b0)253=2353
    Negative Exponent Rulea-n=1an (a0)2-3=123

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

    2,53,x, etc.

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

    23 × 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 pq, where q ≠ 0.

    The general notation of rational exponents is xmn. Here, x is called the base (any real number) and mn 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 23×3, we can now do the following

    23 × 3 = 2312 × 312

    Applying the product to power rule, we obtain

    2312 × 312 = 23×312 = 6912

    Now, coming back to the square root, we obtain

    6912 = 69

    2. Writing the square of a number as a multiplication

    a122=a12·a12

    Now adding the exponents

    a12·a12=a12+12

    Simplifying this, we obtain

    a12+12=a1=a

    Therefore, the square of a12equals to a. Thus, a12=a

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

    a1n and amn.

    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

    a1n=an.

    Write the following in their radical form.

    a13 and4b15

    Solutions

    1. a13=a3

    2. 4b15=4b5

    Express the following in their exponential form.

    x7 and2y

    Solutions

    1. x7=x17

    2. 2y=2y12

    Case 2

    For any positive integer m and n,

    amn=(an)m or amn=amn,

    Write the following in their radical form.

    a23and7b54

    Solutions

    1. a23=a23, which is the same as a23=(a3)2.

    2. 7b54=7b45

    By the Power Rule, we obtain

    7b45=745b45

    Simplifying this further, our final form becomes

    745b45=774b54

    Express the following in their exponential form

    x85and2y83

    Solutions

    1. x85=x85

    2. 2y83=2y38

    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 27-13

    Solution

    By the Negative Exponent Rule,

    27-13=12713

    Now, by the definition of Rational Exponents

    12713=1273

    Simplifying this, we obtain

    1273=1333=13

    Evaluate 6423

    Solution

    By the Power Rule,

    6423=642·13

    Now, with the definition of Rational Exponents

    642·13=6423

    Simplifying this yields

    6423=4323=43·433

    Further tidying up this expression, we have

    43·433=4·4=16

    Real-World Example

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

    r=3V4π13.

    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=3(24)4π13r=724π13r=18π13r=18π3r=1.789400458 units

    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.

    x15·x23

    Solution

    By the Product Rule

    x15·x23=x15+23=x1315

    Simplify the expression below.

    x437

    Solution

    By the Power Rule

    x437=x4·37=x127

    Simplify the following.

    x34x19

    Solution

    By the Quotient Rule

    x34x19=x34-19=x2336

    Simplify the expression below.

    x23y1412

    Solution

    By the Product to Power Rule

    x23y1412=x23·12·y14·12=x13·y18

    Simplify the following

    x12x34x-32y-5413

    Solution

    By the Product Rule

    x12x34x-32y-5413=x12+34x-32y-5413=x54x-32y-5413

    Followed by the Quotient Rule

    x54x-32y-5413=x54--32y-5413=x114y-5413

    Next, by the Product to Power Rule

    x114y-5413=x114·13y-54·13=x1112y-512

    Finally, by the Negative Exponent Rule

    x1112y-512=x11121y512=x1112·y512

    Expressions with Rational Exponents

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

    ConditionExample

    No negative exponents are present

    Instead of writing 32, we should simplify this as 132 by the Negative Exponent Rule

    The denominator is not in the form of a fractional exponent

    Given that 3412, we should express this as 34 by the Definition of Rational Exponents

    It is not a complex fraction

    Rather than writing 532, we can simplify this as 523since 532=5÷32=5×23

    The index of any remaining radical is the least number possible

    Say we have a final result of 32. We can further reduce this by noting that 32=16×2=162=42

    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 pq, where q ≠ 0.
    • Forms of rational exponents
      FormRepresentation
      a1nIf a is a real number andn2a1n=an
      amnFor any positive integer m and namn=anm or amn=amn
    • Properties of exponents
      PropertyDerivation
      Product Ruleam·an=am+n
      Power Rule(am)n=am·n
      Product to Power Ruleabm=ambm
      Quotient Ruleaman=am-n
      Zero Exponent Rulea0=1
      Quotient to Power Ruleabm=ambm
      Negative Exponent Rulea-n=1an
    Rational Exponents Rational Exponents
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    Frequently Asked Questions about Rational Exponents

    What are the properties of rational exponents?

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

    How do you apply properties of rational exponents?

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

    What is the rule for rational exponents?

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

    How do you simplify properties of rational exponents?

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

    Why do we need rational exponents?

    We need rational exponents to solve radical functions

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