Rational Exponents
Rational exponents are another method for writing radicals and can be used to simplify expressions involving both exponents and roots.
Learning Objective

Relate rational exponents to radicals and the rules for manipulating them
Key Points
 If
$b$ is a positive real number and$n$ is a positive integer, then there is exactly one positive real solution to$x^n = b $ . This solution is called the principal$n$ th root of$b$ , denoted$\sqrt[n]{b}$ or$\displaystyle b^{\frac{1}{n}}$ .  A power of a positive real number
$b$ with a rational exponent$\frac{m}{n}$ in lowest terms satisfies${b}^{\frac {m}{n}}= {({b}^{m})}^{\frac{1}{n}}=\sqrt[n]{{b}^{m}}$ .  The rule for multiplying numbers with rational exponents is
$\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}$ .  The rule for dividing numbers with rational exponents is
$\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$ .  Writing an expression in the form
${b}^{\frac {m}{n}}$ can allow you to simplify by cancelling powers and roots.
Terms

root
A number that when raised to a specified power yields the specified number or expression.

rational number
A real number that can be expressed as the ratio of two integers.

exponent
The power raised above the base, representing the number of times the base must be multiplied by itself.
Full Text
A rational exponent is a rational number that provides another method for writing roots. For example, an
There are also cases where the exponent is a fraction
where
The following rules hold true about the signs of roots and rational exponents. For a rational exponent
 The root is positive if
$m$ is even; for example,$(27)^\frac{2}{3}=9$ .  The root is negative for negative
$b$ if$m$ and$n$ are odd; for example,$\displaystyle (27)^\frac{1}{3}=3$ .  The root can be either sign if
$b$ is positive and$n$ is even; for example,$64^\frac{1}{2}$ has two roots:$8$ and$8$ .
Note that since there is no real number
The following are rules for operations on numbers with rational exponents.
Multiplying Numbers with Rational Exponents
The following holds true for any rational exponent:
For example, we can rewrite
Notice that
Dividing Numbers with Rational Exponents
The following holds true for any rational exponent:
For example, we can rewrite
Notice that the denominator can be simplified further:
Therefore, the simplified form is:
Canceling Powers and Roots
In some cases, writing an exponent in its fraction form makes it easier to cancel powers and roots. Recall that
For example, consider
We can simplify the fraction in the exponent to 2, giving us
Example 1
Simplify the following expression:
This expression can be rewritten using the rule for dividing numbers with rational exponents:
Notice that the radical in the denominator is a perfect square and can therefore be rewritten as follows:
Now, notice that the numerator can be rewritten:
Therefore, the simplified form is:
Example 2
Simplify the following expression:
First, rewrite the numerator and denominator in rational exponent form:
Notice that the exponent in the denominator can be simplified:
Recall the rule for dividing numbers with exponents, in which the exponents are subtracted. Applying the division rule, we have:
Thus, the simplified form is simply
Key Term Reference
 denominator
 Appears in these related concepts: Introduction to Rational Functions, Solving Problems with Rational Functions, and Fractions Involving Radicals
 expression
 Appears in these related concepts: Compound Inequalities, Sets of Numbers, and Simplifying, Multiplying, and Dividing
 fraction
 Appears in these related concepts: SI Unit Prefixes, Separable Equations, and Fractions
 imaginary
 Appears in these related concepts: Addition, Subtraction, and Multiplication, Standard Equations of Hyperbolas, and Complex Conjugates and Division
 imaginary number
 Appears in these related concepts: Domains of Rational and Radical Functions, Imaginary Numbers, and Radical Functions
 integer
 Appears in these related concepts: Scientific Notation, Binomial Expansions and Pascal's Triangle, and Finding a Specific Term
 numerator
 Appears in these related concepts: Low Voter Turnout, Rational Equations, and Permutations of Distinguishable Objects
 principal
 Appears in these related concepts: MultiPeriod Investment, Types of Bonds, and Formulas and ProblemSolving
 radical
 Appears in these related concepts: Addition Reactions, Solving Problems with Radicals, and Adding, Subtracting, and Multiplying Radical Expressions
 real number
 Appears in these related concepts: Solving Problems with Inequalities, Introduction to Complex Numbers, and Zeroes of Polynomial Functions with Real Coefficients
 real numbers
 Appears in these related concepts: Piecewise Functions, Introduction to Domain and Range, and Linear Inequalities
 sign
 Appears in these related concepts: Polynomial Inequalities, The Intermediate Value Theorem, and The Rule of Signs
 solution
 Appears in these related concepts: Electrolyte and Nonelectrolyte Solutions, Turning Your Claim Into a Thesis Statement, and What is an Equation?
 square
 Appears in these related concepts: Matrix Multiplication, Factoring a Difference of Squares, and Radical Equations
 term
 Appears in these related concepts: Basics of Graphing Polynomial Functions, The 22nd Amendment, and Introduction to Variables
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