We review the rules of exponents.
Exponents
Recall that the notation means to multiple by itself times. That is,
When we write out all the terms in the product instead of using the exponent notation, we call that expanded form.
The exponent rules that follow below come directly from this definition of exponents.
Product Rule of Exponents
If we write out without using exponents, we’d have: If we then count how many ’s are being multiplied together, we find we have , a total of seven s. So simplifies like this:
This example demonstrates our first exponent rule, the Product Rule:
Power to a Power Rule
If we write out without using exponents, we’d have multiplied by itself: If we again count how many s are being multiplied, we have a total of two groups each with five ’s. So we’d have instances of a . So simplifies like this:
This demonstrates our second exponent rule, the Power to a Power Rule:Product to a Power Rule
The third exponent rule deals with having multiplication inside a set of parentheses and an exponent outside the parentheses. If we write out without using an exponent, we’d have multiplied by itself five times: Keeping in mind that there is multiplication between every and , and multiplication between all of the parentheses pairs, we can reorder and regroup the factors: We could leave it written this way if feels especially large. But if you are able to evaluate , then perhaps a better final version of this expression is .
We essentially applied the outer exponent to each factor inside the parentheses. It is important to see how the exponent applied to both the and the , not just to the .
To simplify , we write this out in its expanded form, as a product of ’s and ’s, we have
Note that the exponent on can simply be applied to both and .
This demonstrates our third exponent rule, the Product to a Power Rule:
Summary of the Rules of Exponents for Multiplication
Product Rule for Exponents Power to a Power Rule Product to a Power Rule
Many examples will make use of more than one exponent rule. In deciding which exponent rule to work with first, it’s important to remember that the order of operations still applies.
Quotient to a Power Rule
One rule we have learned is the Product to a Power Rule, as in . When two factors
are multiplied and the product is raised to a power, we may apply the exponent to
each of those factors individually. We can use the rules of fractions to extend this
property to a quotient raised to a power.
Let be a real number, where .
Find another way to write .
Writing the expression without an exponent and then simplifying, we have:
- (a)
- Simplify .
- (b)
- Simplify . If you end up with a large power of a specific number, leave it written that way.
- (c)
- Simplify . If you end up with a large power of a specific number, leave it written that way.
- (a)
- We can use the Quotient to a Power Rule:
- (b)
- If we stick closely to the order of operations, we should first
simplify inside the parentheses and then work with the outer
exponent. Going this route, we will first use the quotient rule:
Now we can apply the outer exponent to each factor inside the parentheses
using the Product to a Power Rule:
To finish, we need to use the Power to a Power Rule: - (c)
- According to the order of operations, we should simplify inside parentheses first, then apply exponents, then divide. Since we cannot simplify inside the parentheses, we must apply the outer exponents to each factor inside the respective set of parentheses first:
Zero as an Exponent
So far, we have been working with exponents that are natural numbers (). Now, we will expand our understanding to include exponents that are any integer, as with and . As a first step, let’s explore how should behave as an exponent.
Consider for some positive number . What should this be equal to? Well, based on our Product Rule for Exponents, it should be the case that But if that’s the case, the only way can be true is if . (Try dividing both sides by ).
There was nothing special about the that we used except that we said it was positive. In fact, all we really needed was that so that we could divide both sides by it in that last step. Based on this exploration, we make the following definition.
- In the expression , the exponent applies to everything inside the parentheses.
- In the expression the exponent applies to everything inside the parentheses, .
- In contrast to the previous example, if we have , the exponent only applies to the . The exponent has a higher priority than negation in the order of operations: , and so .
- In the expression , the exponent only applies to the :
Negative Exponents
We understand what it means for a variable to have a natural number exponent. For example, means . Now we will try to give meaning to an exponent that is a negative integer, like in .
To consider what it could possibly mean to have a negative integer exponent, let’s extend the pattern for powers of 2 in the table below. In this table, each time we move down a row, we reduce the power by and we divide the value by . We can continue this pattern in the power and value columns, going all the way down to negative exponents.
We see a pattern where is equal to . Note that the choice of base was arbitrary, and this pattern works for all bases except , since we cannot divide by in moving from one row to the next.
Fractional Exponents
So far, all exponents we have addressed are integers. Now we will consider fractional exponents.
We want our fractional exponents to still follow the exponent rules we’ve already defined as well. In particular, we want
It turns out that is not quite enough to always determine a single value for but we do only get one number that will work if we also specify that we want to be a positive, real number. We will explore why we need to state that must be positive later in this course. This leads us to the following definition.
For any nonnegative, real number and any natural number in , we define to mean the number such that and That is, We call this number the th root of and also denote it as
The other exponent rules we have discussed above also apply as expected to fractional powers. In particular, we can use these other exponent rules to define for any integers .
Let’s summarize all the exponent rules that we have discussed in this section.