Properties of Rational Exponents

By: Ashlyn Singer and Cassidy Rupley

Rational Exponents and Radical Notation

To turn a rational exponent into radical form, you place the denominator as the index (outside of the radical) and, if the numerator is greater than 1, you raise everything to that power.

Product of Powers Property

Add exponents. To multiply two powers with the same base, add the exponents.

Quotient of Powers Property

When you multiply two powers with the same base, you add the exponents. (That's the product of powers property.) So when you divide two powers with the same base, you subtract the exponents. In other words, for all real numbers a, b, and c, where a ≠ 0,What you're really doing here is cancelling common factors from the numerator and denominator.

Power of a Power

To find a power of a power, multiply the exponents. This is an extension of the product of powers. Suppose you have a number raised to a power, and you multiply the whole expression by itself over and over. This is the same as raising the expression to a power.

Power of a Product

When you have a power of a product, you raise each factor to the power and then multiply them together.

Power of a Quotient

To solve a power of a quotient, you raise the numerator and the denominator to the power and then divide.