Rational Exponents Square and Cube Roots as Exponents We define a^{1/2} as the nonnegative number such that when you square it, you get a.
Example 9^{1/2} = 3
We define a^{1/3} as the number such that when you cube it you get a.
Example 8^{1/3} = 2
Rational Roots We define a^{1/n} as the unique nonnegative number x such that x^{n} = a If n is odd then the domain of the function f(x) = x^{1/n} is all real numbers and if n is even then the domain of f(x) = x^{1/n} is all nonnegative numbers.
Exercise Which are real numbers:
We define x^{m/n} by (x^{1/n})^{m}
Example: 8^{2/3} = (8^{1/3})^{2} = 2^{2} = 4 In Radical notation the above can be written as:
Rules of Exponents The same basic rules of exponents apply. If you need a review of exponents, go to Rules of Exponents. Or if you want to practice exponents interactively go to Practice Exponents
Example
x^{1/3 }y ^{2/5} = x^{1}y ^{8/20  5/20 } = x y ^{13/20}
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