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 ,