# Properties of Rational Exponents

### By: Michael Migis

We use "radical" notation to disignate roots

## Quotient of powers property

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. Example:

## Product of powers property

If you recall the way exponents are defined, you know that this means:

(7 × 7) × (7 × 7 × 7 × 7 × 7 × 7)

If we take away the parentheses, we have the product of eight 7s, which can be written more simply as:

78

This suggests a shortcut: all we need to do is add the exponents!

72 × 76 = 7(2 + 6) = 78

In general, for all real numbers a, b, and c,

ab × ac = a(b + c)

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

## Power of a Power Property

This property states that if a number that is already raised to certain power is raised again then it is equal to if the two exponents were multiplied by each other. As shown in the picture to the right.

## Power of a Product Property

This property states that if two variables, or a variable multiplied by a number are all raised by an exponent the exponent is distributed to all of the variables/numbers. as shown in the picture to to the right.

## Power of a Quotient Property

If to expressions that are raised to the same exponent are divided by each other then the exponent will remain unchanged and the expressions will be divided as normal. This also applies if there are two espressions being divided by each other that are then raised by an exponent, the exponent is distributed and then the expressions are carried out as normal.