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Rational Exponents
Exponents are shorthand for repeated multiplication and can be used to express an nth root of a number: b is a number x such that x^{n} = b.
Learning Objective

Evaluate exponential expressions of the form , , and
Key Points
 The number in larger font is called the base. The number in superscript (that is, the smaller number written above) is called the exponent.
 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 nth root of b. It is denoted n√b, where √ is the radical symbol; alternatively, it may be written b^{1/n}.
 A power of a positive real number b with a rational exponent m/n in lowest terms satisfies
${b}^{\frac {m}{n}}= {({b}^{m})}^{\frac{1}{n}}=\sqrt[n]{{b}^{m}}$ .
Term

rational number
A real number that can be expressed as the ratio of two integers.
Full Text
Exponents are a shorthand used for repeated multiplication. Remember that when you were first introduced to multiplication it was as a shorthand for repeated addition. For example, you learned that:
For example:
Here are some other examples:
Rational Exponents
A rational exponent is a rational number that can be used as another way to write roots. An nth root of a number b is a number x such that x^{n} = b.
If b is a positive real number and n is a positive integer, then there is exactly one positive real solution to
When one speaks of the nth root of a positive real number b, one usually means the principal nth root.
where m is an integer and n is a positive integer.
Rational powers m/n, where m/n is in lowest terms, are positive if m is even, negative for negative b if m and n are odd, and can be either sign if b is positive and n is even.
 If n is even, then x^{n} = b has two real solutions;
 if b is positive, which are the positive and negative nth roots.
 The equation has no solution in real numbers if b is negative.
 If n is odd, then x^{n} = b has one real solution.
 The solution is positive if b is positive and negative if b is negative.
Examples of exponents graphed can be seen in this figure .
Interactive Graph: Exponential graph with different substitute "b" values
Graphs of
Key Term Reference
 base
 Appears in these related concepts: Logarithms of Powers, The Role of the Kidneys in AcidBase Balance, and Biology: DNA Structure and Replication
 equation
 Appears in these related concepts: A General Approach, Equations and Inequalities, and Equations and Their Solutions
 exponent
 Appears in these related concepts: Integer Exponents, Logarithms of Products, and Scientific Notation
 expression
 Appears in these related concepts: Simplifying, Multiplying, and Dividing, Expressions and Sets of Numbers, and Bacterial Transformation
 imaginary
 Appears in these related concepts: The Fundamental Theorem of Algebra, The Quadratic Formula, and Complex Conjugates and Division
 imaginary unit
 Appears in this related concept: Addition, Subtraction, and Multiplication
 integer
 Appears in these related concepts: Exact Numbers, Scientific Notation, and Finding a Specific Term
 principal
 Appears in these related concepts: MultiPeriod Investment, Types of Bonds, and Maximizing Shareholder and Market Value
 radical
 Appears in these related concepts: Reactions of Alkanes, Radical Functions, and Solving Problems with Radicals
 real number
 Appears in these related concepts: Graphing the Normal Distribution, Intermediate Value Theorem, and The ComplexNumber System
 real numbers
 Appears in these related concepts: Finding Domains of Functions, Zeroes of Polynomial Functions with Real Coefficients, and Linear Inequalities
 root
 Appears in these related concepts: Newton's Method, The Rule of Signs, and Radical Equations
 sign
 Appears in these related concepts: The Intermediate Value Theorem, Sequences of Statements, and Scientific Notation
 term
 Appears in these related concepts: Arithmetic Sequences, Basics of Graphing Polynomial Functions, and The 22nd Amendment
Sources
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Cite This Source
Source: Boundless. “Rational Exponents.” Boundless Algebra. Boundless, 21 Jul. 2015. Retrieved 22 Jul. 2015 from https://www.boundless.com/algebra/textbooks/boundlessalgebratextbook/thebuildingblocksofalgebra1/radicalnotationandexponents14/rationalexponents8511070/