Watch
Watching this resources will notify you when proposed changes or new versions are created so you can keep track of improvements that have been made.
Favorite
Favoriting this resource allows you to save it in the “My Resources” tab of your account. There, you can easily access this resource later when you’re ready to customize it or assign it to your students.
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 .
Assign just this concept or entire chapters to your class for free.
Key Term Reference
 base
 Appears in these related concepts: Strong Bases, The Role of the Kidneys in AcidBase Balance, and Biology: DNA Structure and Replication
 equation
 Appears in these related concepts: Motion with Constant Acceleration, A General Approach, and Equations and Inequalities
 exponent
 Appears in these related concepts: Logarithms of Powers, Scientific Notation, and Logarithms of Quotients
 expression
 Appears in these related concepts: Expressions and Sets of Numbers, Bacterial Transformation, and Equations and Their Solutions
 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: Types of Bonds, Maximizing Shareholder and Market Value, and Formulas and ProblemSolving
 radical
 Appears in these related concepts: Addition Reactions, Solving Problems with Radicals, and Adding, Subtracting, and Multiplying
 real number
 Appears in these related concepts: Intermediate Value Theorem, Solving Problems with Inequalities, 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: Radical Functions, 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
Boundless vets and curates highquality, openly licensed content from around the Internet. This particular resource used the following sources:
Cite This Source
Source: Boundless. “Rational Exponents.” Boundless Algebra. Boundless, 21 Jul. 2015. Retrieved 08 Feb. 2016 from https://www.boundless.com/algebra/textbooks/boundlessalgebratextbook/thebuildingblocksofalgebra1/radicalnotationandexponents14/rationalexponents8511070/