Explainer: Dividing Monomials

In this explainer, we will learn how to divide monomials involving single and multiple variables.

Consider the expression 4.

The 4 is called the base and the little 5 is called the exponent. The exponent tells us the number of times that the base (or number 4) has been multiplied by itself. In this case, 4=4×4×4×4×4.Five4smultipliedtogether

If we evaluate this expression, we get 1,024.

Often in mathematics, we are asked to simplify expressions involving exponents but are not always required to evaluate them. This is particularly true with large exponents where the calculation would be long and cumbersome and it is sufficient to leave our answer in a form containing exponents. Let us look now at how to simplify an expression that involves a quotient of two exponential expressions. An example of this would be 55.

If we write the top and bottom of the expression in expanded form, we get 5×5×5×5×5×55×5×5.

Recall that dividing the top and bottom of a fraction by a number does not change its value. We can, therefore, divide the top and bottom of this expression by 5, three times. This gives us 5×5×5×5×5×55×5×5, which is the same as 5.

Effectively, when simplifying the quotient by dividing through by 5, three times, we have reduced each exponent by 3. Therefore, more generally, if we have a quotient 𝑥𝑥, provided the two exponential terms have the same base, this will always simplify to 𝑥.

This is our general rule that we can highlight now.

Key Rule: The Quotient Rule of Exponents

We have 𝑥𝑥=𝑥.

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

Let us look at some examples.

Example 1: Introducing the Quotient Rule of Exponents

Simplify 𝑥÷𝑥.

Answer

If we start by recalling the quotient rule of exponents, 𝑥÷𝑥=𝑥, we can then use this to simplify the expression. In this case, 𝑎=6 and 𝑏=4, which tells us that our expression can be simplified as follows: 𝑥÷𝑥=𝑥=𝑥.

We can also solve this problem from first principles. Remember that 𝑥÷𝑥=𝑥𝑥.

If we then expand out the top and the bottom of the quotient, we get 𝑥×𝑥×𝑥×𝑥×𝑥×𝑥𝑥×𝑥×𝑥×𝑥.

We can then divide the top and bottom through by 𝑥 four times, giving us 𝑥×𝑥×𝑥×𝑥×𝑥×𝑥𝑥×𝑥×𝑥×𝑥, which simplifies to 𝑥.

Example 2: Using the Quotient Rule of Exponents to Simplify an Expression

Simplify 44.

Answer

If we start by recalling the quotient rule of exponents, 𝑥𝑥=𝑥, we can then use this to simplify the expression. In this case, the quotient contains two exponential expressions that have the same base, so our rule can be used. To simplify the expression, we just have to subtract the exponent on the bottom from the exponent on the top. That is, 44=4=4.

As the question says to simplify the expression, not evaluate, we are not expected to calculate the numerical value. This, after all, would be very large!

Questions that require the use of the quotient rule of exponents often also require the use of the product rule of exponents in order to fully simplify. Let us recall the product rule of exponents.

Key Rule: Products of Exponential Expressions with the Same Base

We have 𝑥×𝑥=𝑥.

When answering questions that contain both products and quotients, it is usually best to simplify any products first and then simplify any resulting quotients. Let us look at a couple of examples.

Example 3: Using the Product and Quotient Rules of Exponents to Simplify Expressions

Simplify 𝑥×𝑥𝑥, where 𝑥0.

Answer

Let us start by recalling the product rule of exponents, 𝑥×𝑥=𝑥, and the quotient rule of exponents, 𝑥𝑥=𝑥.

If we focus first on the expression at the top of the quotient, 𝑥×𝑥, we can simplify this using the product rule to get 𝑥=𝑥.

We can then rewrite our quotient as 𝑥𝑥.

If we then apply the quotient rule, we can simplify this to 𝑥=𝑥.

Example 4: The Product and Quotient Rules of Exponents

Simplify 𝑥𝑦×𝑥𝑦𝑥𝑦.

Answer

Let us start by recalling the product rule of exponents, 𝑥×𝑥=𝑥, and the quotient rule of exponents, 𝑥𝑥=𝑥.

If we focus on the top of the quotient, we can rewrite this as 𝑥×𝑦×𝑥×𝑦.

This can then be rearranged to 𝑥×𝑥×𝑦×𝑦; then it can be simplified using the product rule to get 𝑥×𝑦.

If we then rewrite this as a quotient, we get 𝑥×𝑦𝑥×𝑦.

Finally, simplifying each of the variables separately using the quotient rule, we get an answer of 𝑥𝑦.

A particular issue that we have not yet addressed is what happens if the exponent on the bottom of a quotient is larger than the exponent on the top. For example, 𝑥𝑥.

If we expand the top and bottom of the quotient, we get 𝑥×𝑥×𝑥×𝑥𝑥×𝑥×𝑥×𝑥×𝑥×𝑥.

At this point, we can see that we can divide the top and the bottom through by 𝑥 four times to get 𝑥×𝑥×𝑥×𝑥𝑥×𝑥×𝑥×𝑥×𝑥×𝑥, which is equal to 1𝑥.

Alternatively, if we simplify the original expression with the quotient rule of exponents, we get 𝑥=𝑥.

Both of these answers must be correct, which means that they are equivalent forms. That is, 𝑥=1𝑥.

This is, in fact, a first introduction to negative exponents. In general, 𝑥=1𝑥.

Let us finish by having a look at one final question.

Example 5: The Quotient Rule of Exponents

Simplify 𝑥𝑥.

Answer

If we start by recalling the quotient rule of exponents, 𝑥𝑥=𝑥, we can then use this to simplify the expression. In this case, the quotient contains two exponential expressions that have the same base, so our rule can be used. To simplify the expression, we just have to subtract the exponent on the bottom from the exponent on the top. That is, 𝑥𝑥=𝑥=𝑥.

If we solved this by expanding the two exponential terms and then canceling from the top and bottom of the exponent, we would see that the solution could also be written as 1𝑥, and this is an equivalent solution.

Key Points

  1. If you have a quotient of two exponential expressions that have the same base, then we can use the quotient rule of exponents to simplify the expression 𝑥𝑥=𝑥.
    Note that this can also be written as 𝑥÷𝑥=𝑥. For example, 2÷2=2=2.

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