91Ó°ÊÓ

When dividing monomials, one of the first steps is to simplify the coefficients. Coefficients are the numerical part of each term. To simplify the coefficients, you divide them like any regular numbers.

In our example, we have the coefficients 8 and 64. To simplify, we divide 8 by 64.

\[\frac{8}{64} = \frac{1}{8}\]

This means the simplified coefficient of our result is \(\frac{1}{8}\). Simplifying coefficients helps break down the expression and makes it easier to deal with the variables.

Next, we need to simplify the variables. When the same variable appears in both the numerator and the denominator, you can often cancel them out.

Let's look at the variable \(m\). In our example, \(m\) is present in both the numerator and the denominator:
\(\frac{m}{m} = 1\)

Since \(m\) divided by \(m\) equals 1, they cancel out, and we are left without any \(m\) in our expression.

Now, let's move on to the variable \(n\). The variable \(n\) is raised to powers in both the numerator and the denominator: \(\frac{n^{10}}{n^4}\). To simplify, subtract the exponent in the denominator from the exponent in the numerator:

\[\frac{n^{10}}{n^4} = n^{10-4} = n^6\]

So, the variable \(n\) simplifies to \(n^6\).

When dealing with powers and exponents in division, there are a few rules to keep in mind. One important rule is:

• When you divide terms with the same base, subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).

This rule helps simplify expressions with exponents easily.

In our example, we applied this rule to the variable \(n\): \[\frac{n^{10}}{n^4} = n^{10-4} = n^6\]

This means that you simply subtract 4 from 10 to get 6.

Finally, after simplifying both the coefficients and the variables, we combine all the simplified parts together. Our fully simplified expression is: \[\frac{1}{8} \times 1 \times n^6 = \frac{n^6}{8}\].

Understanding these rules and steps will help you tackle any similar problems with confidence.