In this explainer, we will learn how to simplify monomials with negative exponents.
Let us start by recalling what we know about positive exponents. For an expression , this means
Here, the exponent, 4, is informing us of the number of times 2 has been multiplied by itself. Equally, if we consider the expression , if is a positive integer, the is telling us the number of times is multiplied by itself. What then happens if we have a negative exponent, such as ?
How do we multiply a number by itself times? This, of course, is not a sensible way of introducing negative exponents. Let us start by introducing a pattern of reducing powers of 2:
Looking at this pattern, it looks like each reduction of the exponent, of 2, by one results in the number on the right-hand-side being divided by 2. If we were to continue this pattern, it would be reasonable to assume that an exponent of (that is, ) would be equal to , which is . Similarly, an exponent of would be , which equals . We could continue the list as follows:
Is our assumption correct? Let us see if we can justify this by using the quotient rule for exponents. Recall that
If we consider the expression
using the quotient rule of exponents, we can simplify this to
Equally, we can rewrite the expression as
by expanding the exponents. We can then divide the top and bottom by 2 three times, which gives us
This simplifies to
which can be evaluated to
If we refer back to our list of exponents of 2, we can see that our assumption, at least for the value of , is indeed correct. We could take a very similar approach to show that all of our list of powers of 2 was correct. One point to highlight in the above construction is the fact that we showed
This can be generalized for any nonzero raised to a negative exponent.
Key Information: Negative Exponents
For any nonzero , we have that
This is, in fact, a special case of the quotient rule of exponents. Recall again that
In the above generalization, and . We have, therefore, that
As , for any nonzero , we can simplify this to
Before looking at a few examples, let us state all the rules of exponents that are often used in combination with what we have just discovered about negative exponents. We have the product rule, the quotient rule, the power rule, and the zero exponent property.
Key Information: Exponent Rules
- Product rule of exponents:
- Quotient rule of exponents:
- Power rule of exponents:
- Zero exponents:
Example 1: Evaluating a Number Raised to a Negative Exponent
Evaluate .
Answer
Recall that for any nonzero , we have that
Using this, we can rewrite the expression to be
We can now evaluate , which equals 196, to find that the original expression is equal to
Example 2: Evaluating Expressions Containing Negative Exponents
Evaluate .
Answer
Recall that for any nonzero , we have that
and that
Using the quotient rule of exponents, we can rewrite our expression as follows:
Using the negative exponent rule, we can rewrite this as
At this point, we can evaluate the expression. Using the fact that , we have a final solution of
Example 3: Recognizing Equivalent Expressions Involving Negative Exponents
Which of the following is equivalent to ?
- 27
Answer
Here, we have a number raised to the power of . A common mistake would be to evaluate this to , which is the product of the two numbers. This, however, is not correct. We need to recall the negative exponent rule: for any nonzero we have that
We can use this to rewrite our expression:
This, however, is not the same as any of the options. We need to evaluate the denominator of the expression:
Our answer is, therefore, D.
All of the rules that we have been using for the previous examples can be applied to both numerical and algebraic expressions. Let us now move on to look at some algebraic examples.
Example 4: Evaluating Algebraic Expressions Containing Negative Exponents
Simplify .
Answer
To answer this question, let us first recall the power rule of exponents:
If we distribute the power across each of the terms and apply the power rule of exponents, we get
Simplifying each term gives us
We can then reorder the two terms:
At this point, you can see that we can multiply together the two numbers and use the product rule of exponents to simplify the terms. This gives us
Simplifying, we find that our answer is
This is our answer. However, it is worth noting that we can present this in a different form that would be equally valid. If we recall the rule for negative exponents, which states that for any nonzero ,
we can rewrite the as . This means that our solution can be written as
which is the same as
We will finish by looking at one final example that combines multiple exponent rules.
Example 5: Combining Exponent Rules to Simplify Expressions Containing Negative Exponents
Simplify , given that .
Answer
Let us start by recalling the product and power rules of exponents:
- Product rule of exponents:
- Power rule of exponents:
Using the power rule first, we can rewrite the top of the rational expression as follows:
Simplifying, we get
If we now use the product rule of exponents, we rewrite the top and the bottom of our expression to give us
Simplifying, we get
At this stage, let us recall the quotient rule of exponents:
Using this, we can rewrite the expression as follows:
Being very careful with negatives and remembering that subtracting a negative is the same as adding, we can simplify the expression to give us
This is our answer, but it is worth noting that this can be rewritten using the rule for negative exponents. Remember that for any nonzero , we have that
This means that we can rewrite our answer as
Key Points
- Negative exponents are defined as follows: for any nonzero , we have that
- When working with negative exponents, we may need to use the following rules:
- Product rule of exponents:
- Quotient rule of exponents:
- Power rule of exponents:
- Zero exponents: for any nonzero ,