# Exponents

## Fractional Exponents

Sometimes exponents are written as fractions.  Fractional exponents contain two pieces of information:

• The exponent that the base should be raised to (the numerator of the fraction).
• The index of the root that that should be applied to the base and exponent (the denominator of the fraction).

For example, here are a set of fractional exponents written as radicals:

## Exponents (Negative Exponents)

Exponents (sometimes called powers) are used when you want to multiply a number (or variable) times itself a certain number of times.  Technically, because anything raised to the power of 0 is one (e.g., $8^0=1$), when you raise a number to a higher power, you are multiplying 1 by the base number as many times as the exponent tells you to.

\eqalign{8^0&=1\\8^1&=1 \times 8 = 8\\8^2&=1 \times8\times8=64\\8^3&=1\times8\times8\times8=512}

You see the pattern.

## Exponents (Exponents of Exponents)

Sometimes when working with exponents, you'll see exponents raised to another number, such as: $(x^2)^5$

The rules of exponents still apply.  The exponent of 5, means that you want to raise the base $(x^2)$ to the power of 5. In other words, multiply $(x^2)$ times itself 5 times.

## Dividing Exponents (Same Base)

You have probably already learned that when you multiply exponents with the same base, you can just add the exponents (if you haven't, check out the detailed Multiplying Exponents lesson).

Division with exponents, as long as they have the same base, follows the same principle.  If the numbers have the same base, you can subtract the exponents for the final answer.

## Multiplying Exponents (same base)

When you get into algebra, you'll often multiply terms that already have exponents in them.

When you multiply exponents with the same base, you can bring those exponents together into one term  -- and there are some good tricks for doing it quickly and efficiently.

$\text{Base}\rightarrow \large{17}^{\large{3}\leftarrow\text{exponent}}$