Explainer: Simplifying Monomials: Zero Exponents

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.

Key Information

The Quotient Rule of Exponents: 𝑥𝑥=𝑥.

Note that this can also be written as 𝑥÷𝑥=𝑥.

Let us start by considering the expression 22.

Using the quotient rule of exponents, it can be rewritten as 2, which simplifies to 2.

So, what does this mean? If we look again at the original expression, we can evaluate the top and bottom to get 88, which simplifies to give us an answer of 1.

Now, these two things must be equivalent, which means 2=1.

This, in fact, can be generalized for any expression in this form. That is, for nonzero 𝑥, 𝑥𝑥 simplifies to 𝑥.

Equally, if we divide 𝑥 by 𝑥, we must also get 1. This means that, for any 𝑥, 𝑥=1.

Let us state this formally.

Key Information: The Zero Exponent

For any nonzero variable 𝑥, 𝑥=1.

An alternative approach that may help you visualize this is to consider the following pattern of powers: 2=16,2=8,2=4,2=2,2=1.

To go from one line to the next, we need to divide by 2. With this construction, we can see that 2=2 and then 2=1.

We will now look at a few examples.

Example 1: Raising a Number to the Power of Zero

Evaluate 12.

Answer

Recall that, for any nonzero 𝑥, 𝑥=1. In this case, we have 𝑥=12. Therefore, 12=1.

Another way of approaching this is to note that 12=12.

If we recall the quotient rule of exponents, which states that 𝑥𝑥=𝑥,

we can rewrite 12

as 1212,

which simplifies to 1.

Example 2: Raising a Monomial to the Power of Zero

Determine the value of (12𝑎), given that 𝑎0.

Answer

We can start by distributing the power over the monomial. That is, (12𝑎)=12𝑎.

Now, recall that, for any nonzero 𝑥, 𝑥=1.

We have then that 12=1 and 𝑎=1. This means that our original expression simplifies to 1. That is, (12𝑎)=1.

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

Simplify 13𝑥.

Answer

In this question, only the variable 𝑥 is being raised to the power zero, not the term “13𝑥.” The expression is the same as 13×𝑥.

Recall that any nonzero variable raised to the power of zero is equal to one. This means that the expression can be rewritten as 13×1,

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

Simplify 0.314.

Answer

Here, we have a single number raised to the power of zero. Recall that, for any nonzero 𝑥, 𝑥=1. In this case, as 0.314 is nonzero, we have that 0.314=1.

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 0.31×4, 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

Simplify 𝑥𝑥.

Answer

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 𝑥𝑥=𝑥,

we can then use this to simplify the expression. That is, 𝑥𝑥=𝑥=𝑥.

We know that any nonzero variable raised to the power of zero equals one, meaning our expression 𝑥𝑥

simplifies to 1. You may equally have noticed this directly: we have a term being divided by itself, which means that, providing 𝑥 is nonzero, the expression simplifies to 1. This highlights why 𝑥 is, in fact, equal to 1.

Key Points

  1. For any nonzero variable 𝑥, 𝑥=1.
  2. The quotient rule of exponents tells us that, for nonzero 𝑥, 𝑥𝑥=𝑥. In particular, 𝑥𝑥=𝑥=𝑥=1.
  3. 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.

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