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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 .
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Key Term Reference
 base
 Appears in this related concepts: Logarithms of Powers, Balancing Redox Equations, and Strong Bases
 equation
 Appears in this related concepts: Centripetial Acceleration, A General Approach, and Equations and Inequalities
 exponent
 Appears in this related concepts: Logarithmic Functions, Logarithmic Functions, and Simplifying Expressions of the Form log_a a^x and a(log_a x)
 expression
 Appears in this related concepts: Simplifying, Multiplying, and Dividing, Compound Inequalities, and Equations and Their Solutions
 imaginary
 Appears in this related concepts: Addition, Subtraction, and Multiplication, The Fundamental Theorem of Algebra, and Complex Conjugates and Division
 integer
 Appears in this related concepts: Scientific Notation, Total Number of Subsets, and Finding a Specific Term
 principal
 Appears in this related concepts: Types of Bonds, Defining Agency Conflicts, and Maximizing Shareholder and Market Value
 radical
 Appears in this related concepts: Addition Reactions, Reactions of Alkanes, and Adding, Subtracting, and Multiplying
 real number
 Appears in this related concepts: Graphing the Normal Distribution, Solving Problems with Inequalities, and The ComplexNumber System
 real numbers
 Appears in this related concepts: Piecewise Functions, Linear Inequalities, and Factoring Trinomials of the Form ax^2 + bx + c; Perfect Squares
 root
 Appears in this related concepts: Radical Functions, The Rule of Signs, and Radical Equations
 sign
 Appears in this related concepts: The Intermediate Value Theorem, Sequences of Statements, and Polynomial Inequalities
 term
 Appears in this related concepts: Basics of Graphing Polynomial Functions, The 22nd Amendment, and Democracy
Sources
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Cite This Source
Source: Boundless. “Rational Exponents.” Boundless Algebra. Boundless, 03 Jul. 2014. Retrieved 24 Apr. 2015 from https://www.boundless.com/algebra/textbooks/boundlessalgebratextbook/thebuildingblocksofalgebra1/radicalnotationandexponents14/rationalexponents8511070/