Exponent Rules

The basic rules for working with exponents are listed below.

Now, we will see how these rules are applied by working through several examples
where we simplify expressions involving exponents.

Example 1 Simplify
Using Rule 4, we add the exponents . So, we have:

Example 2 Simplify
Using Rule 5, we subtract the exponents . So, we have:

Now, by Rule 2, we know that a negative exponent means that the power is really in
the denominator , so we have:

Example 3 Simplify
Using Rule 7, we can distribute the power through the parentheses. So, we have:

Now, by Rule 6, we know that we multiply exponents together when one exponent is
raised to another exponent. Now, we have:

Example 4 Simplify

Using Rule 3, we can rewrite the square root as an exponent . In a fractional
exponent, the top number re presents the power and the bottom number represents the
root, so the square root would be rewritten as the one-half power:

Now, using Rule 8, we can distribute the power through the parentheses, and, by using
Rule 6, we will multiply the exponents together:

Example 5 Simplify
Using Rule 3, we know that the fractional exponent represents raising 64 to the
third power, since the numerator is 3, and taking the fourth root, since the denomi-
nator is 4. If it will work out evenly, start by taking the root first, this al lows you to
work with smaller numbers. So, we have:

Example 6 Simplify
There are no rules that allow us to bring the square into the parentheses when
they contain a plus or minus. In this case, we need to multiply the power out by using
the "FOIL" method . "FOIL" means to multiply the First, Outside, Inside, and Last
terms together . So, doing this and using Rule 4 to simplify the terms, we have:

Example 7 Simplify

Start by simplifying insided the parentheses. Using Rule 1, we recognize that w⁰ =
1; therefore, we can ignore that factor. Now, use Rule 5 to combine the factors in the
numerator with the factors in the denominator: