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Rational Exponents
Rational exponents are another way to write radicals, and can be applied to simplifying expressions that include 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$b$ . It is denoted$\sqrt[n]{b}$ , where$\sqrt{}$ is the radical symbol; alternatively, it may be written$\displaystyle b^{\frac{1}{n}}$ .  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 which, 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
In an exponential expression, this is the value that is raised above the base, and represents the number of times the base must be multiplied by itself.
Full Text
Rational Exponents
A rational exponent is a rational number that can be used as another way to write roots. For example, an
Rational exponents
There are also cases where the exponent is a fraction
where
For a rational exponent
Note that since there is no real number
The following are rules for operations on numbers with rational exponents that make simplifying them straightforward.
Multiplying Numbers with Rational Exponents
The following holds true for any rational exponents:
For example, we can rewrite
Dividing Numbers with Rational Exponents
The following holds true for any rational exponents:
For example, we can rewrite
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
Example
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 be rewritten:
Now, notice that the numerator can be rewritten:
Therefore, the simplified form is
Example
Simplify the following expression:
First, rewrite the numerator and denominator in rational exponent form:
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
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Key Term Reference
 denominator
 Appears in these related concepts: Complex Conjugates and Division, Rational Equations, and Fractions Involving Radicals
 expression
 Appears in these related concepts: Compound Inequalities, Sets of Numbers, and Graphs of Equations as Graphs of Solutions
 fraction
 Appears in these related concepts: SI Unit Prefixes, Separable Equations, and Fractions
 imaginary
 Appears in these related concepts: The Fundamental Theorem of Algebra, The Quadratic Formula, and Standard Equations of Hyperbolas
 imaginary unit
 Appears in these related concepts: Addition, Subtraction, and Multiplication, Multiplication of Complex Numbers, and Complex Numbers and the Binomial Theorem
 integer
 Appears in these related concepts: Scientific Notation, Finding a Specific Term, and Binomial Expansions and Pascal's Triangle
 numerator
 Appears in these related concepts: Low Voter Turnout, Solving Problems with Rational Functions, 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: Radical Functions, Solving Problems with Radicals, and Adding, Subtracting, and Multiplying Radical Expressions
 real number
 Appears in these related concepts: Intermediate Value Theorem, Solving Problems with Inequalities, and Introduction to Complex Numbers
 real numbers
 Appears in these related concepts: Introduction to Domain and Range, Piecewise Functions, and Linear Inequalities
 sign
 Appears in these related concepts: The Intermediate Value Theorem, The Rule of Signs, and Polynomial Inequalities
 square
 Appears in these related concepts: Factoring a Difference of Squares, Radical Equations, and Matrix Multiplication
 term
 Appears in these related concepts: Basics of Graphing Polynomial Functions, The 22nd Amendment, and Introduction to Variables
Sources
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Cite This Source
Source: Boundless. “Rational Exponents.” Boundless Algebra. Boundless, 22 Jun. 2016. Retrieved 27 Jul. 2016 from https://www.boundless.com/algebra/textbooks/boundlessalgebratextbook/numbersandoperations1/furtherexponents13/rationalexponents8511070/