In this explainer, we will learn how to simplify monomials with an exponent of zero, as any nonzero number raised to the zero power is equal to one.
Let us start by recalling the quotient rule for exponents, which will be very useful when thinking about expressions raised to an exponent of zero.
The Quotient Rule of Exponents:
Note that this can also be written as
Let us start by considering the expression .
Using the quotient rule of exponents, it can be rewritten as , which simplifies to .
So, what does this mean? If we look again at the original expression, we can evaluate the top and bottom to get , which simplifies to give us an answer of 1.
Now, these two things must be equivalent, which means .
This, in fact, can be generalized for any expression in this form. That is, for nonzero , simplifies to .
Equally, if we divide by , we must also get 1. This means that, for any ,
Let us state this formally.
Key Information: The Zero Exponent
For any nonzero variable ,
An alternative approach that may help you visualize this is to consider the following pattern of powers:
To go from one line to the next, we need to divide by 2. With this construction, we can see that and then .
We will now look at a few examples.
Example 1: Raising a Number to the Power of Zero
Recall that, for any nonzero , . In this case, we have . Therefore,
Another way of approaching this is to note that
If we recall the quotient rule of exponents, which states that
we can rewrite
which simplifies to 1.
Example 2: Raising a Monomial to the Power of Zero
Determine the value of , given that .
We can start by distributing the power over the monomial. That is,
Now, recall that, for any nonzero ,
We have then that and . This means that our original expression simplifies to 1. That is,
In this second example, the whole monomial is being raised to the power of zero. We may also face questions where this is not the case. Let us look at an example now.
Example 3: Simplifying a Monomial Where a Variable Is Raised to the Power of Zero
In this question, only the variable is being raised to the power zero, not the term “.” The expression is the same as
Recall that any nonzero variable raised to the power of zero is equal to one. This means that the expression can be rewritten as
which equals 13.
Let us finish by looking at a couple more examples where we use zero powers to simplify expressions.
Example 4: Simplifying a Number Raised to a Zero Power
Here, we have a single number raised to the power of zero. Recall that, for any nonzero , . In this case, as 0.314 is nonzero, we have that
A common mistake when raising a number to the power of zero is to confuse this with the number multiplied by zero and writing the answer as “0.” Another common mistake that could be made with the previous example would be to interpret it as , which is also not correct. Always be careful to spend time thinking about the exponent rules that you know and use these to help you answer the question.
Example 5: Simplifying an Expression Using the Zero Exponent
With this question we can take two approaches to find the solution, both of which are intrinsically related. Firstly, if we recall the quotient rule of exponents, which states that
we can then use this to simplify the expression. That is,
We know that any nonzero variable raised to the power of zero equals one, meaning our expression
simplifies to 1. You may equally have noticed this directly: we have a term being divided by itself, which means that, providing is nonzero, the expression simplifies to 1. This highlights why is, in fact, equal to 1.
- For any nonzero variable ,
- The quotient rule of exponents tells us that, for nonzero , In particular,
- A common mistake is confusing raising to the power of zero with multiplication by zero. Raising to the power of zero gives an answer of one, whereas multiplication by zero gives and answer of zero.