Until this point we have only had exponents that are integers (positive or negative whole numbers), so it is time to introduce two new rules that deal with rational (or fractional) exponents. These rules will help to simplify radicals with different indices by rewriting the problem with rational exponents. Here are the new rules along with an example or two of how to apply each rule:

 The Definition of : , this says that if the exponent is a fraction, then the problem can be rewritten using radicals. Notice that the denominator of the fraction becomes the index of the radical. The Definition of : , this says that if the exponent is a fraction, then the problem can be rewritten using radicals. Notice that the denominator of the fraction becomes the index of the radical and the numerator becomes the power inside the radical.

Example 1 – Rewrite using radicals.

 Step 1: Apply the definition of .

Example 2 – Rewrite using rational exponents.

 Step 1: Apply the definition of .

Example 3 – Rewrite using radicals.

 Step 1: Apply the definition of .

Example 4 – Rewrite using rational exponents.

 Step 1: Apply the definition of .

Below is a complete list of rule for exponents along with a few examples of each rule:

 Zero-Exponent Rule: a0 = 1, this says that anything raised to the zero power is 1. Power Rule (Powers to Powers): (am)n = amn, this says that to raise a power to a power you need to multiply the exponents. There are several other rules that go along with the power rule, such as the product-to-powers rule and the quotient-to-powers rule. Negative Exponent Rule: , this says that negative exponents in the numerator get moved to the denominator and become positive exponents. Negative exponents in the denominator get moved to the numerator and become positive exponents. Only move the negative exponents. Product Rule: am ∙ an = am + n, this says that to multiply two exponents with the same base, you keep the base and add the powers. Quotient Rule: , this says that to divide two exponents with the same base, you keep the base and subtract the powers. This is similar to reducing fractions; when you subtract the powers put the answer in the numerator or denominator depending on where the higher power was located. If the higher power is in the denominator, put the difference in the denominator and vice versa, this will help avoid negative exponents. The Definition of : , this says that if the exponent is a fraction, then the problem can be rewritten using radicals. Notice that the denominator of the fraction becomes the index of the radical. The Definition of : , this says that if the exponent is a fraction, then the problem can be rewritten using radicals. Notice that the denominator of the fraction becomes the index of the radical and the numerator becomes the power inside the radical.
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