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# Properties of Exponents I

## A. Product Rule

Notice the following: Similarly, In general, Thus, when we multiply powers , we add exponents .

Example:

Find  (by adding exponents)

## B. Quotient Rule

Notice the following: Similarly, In general, Thus, when we divide powers , we subtract exponents .

Example:
Find  (by subtracting exponents )

## C. Power Rule

Notice the following: Similarly, In general, Thus, when we take a power of a power , we multiply exponents.

Example:

Find  (by multiplying exponents)

WARNING: Don’t get the product and power rules confused: ## D. Zero Power

Notice the following: , but by the quotient rule, Thus, we see that Similarly, The above argument works for any number , except zero, because we would have which is indeterminate.

In general, ## E. Negative Exponents

Notice the following: However by the quotient rule.

Thus, Similarly, In general, Thus, a negative exponent means reciprocal.

Example:
Find  (negative exponent rule)

## F. Switch Rule

By using the reciprocal idea for negative exponents, we get the switch rule: Thus, in a fraction, powers in the top get sent to the bottom by negative exponents, and
powers in the bottom get sent to the top by negative exponents.

Example:
Rewrite without negative exponents:  (using the switch rule)

## G. Multiple Power Rule

By using the power rule idea , we see that if we have a fraction to a power , we hit every
entry with the power (and multiply exponents).

Thus, Example:
Find  1. It is important to memorize these rules, and to not get them confused.

2. Parentheses are important !: 3. If everything “leaves” a numerator or denominator , you leave “1” behind:

For example: 4. With fractions, variables not in a fraction are considered to be in the numerator.

For example: is the same as 5. When using the quotient rule:

If you use it on a variable, the answer goes in the numerator. If you don’t use it, the
variable stays put.

For example: In this example, are put in the numerator, and w stays put.

6. When using the multiple power rule:

It is only correct to use it if the variables are not added or subtracted: can’t be simplified by this rule .

We will explain how to simplify later in the course. For the time being,
remember the rule:

DON’T APPLY POWERS ACROSS PLUS OR MINUS SIGNS

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