91Ó°ÊÓ

Negative exponents might seem tricky at first, but they follow straightforward rules. For any non-zero base, a negative exponent means you take the reciprocal of the base with a positive exponent. For instance, \(a^{-n} = \frac{1}{a^n}\). This means that instead of multiplying the base several times, you divide 1 by the base raised to the positive exponent.
Let's apply this to our example: \(m^{-5}\) becomes \(\frac{1}{m^5}\). This helps transform complex algebraic fractions into more manageable forms.
To convert expressions with negative exponents, follow these steps:

• Identify the parts with negative exponents.
• Rewrite the negative exponents as reciprocals with positive exponents.
• Simplify the resulting expression if necessary.

By following these steps, you can steer clear of confusion and keep your algebraic fractions simplified and easier to understand.

Simplifying coefficients in an algebraic fraction means reducing the numerical parts in the fraction to their simplest form. Suppose we have \(\frac{15}{10}\). This fraction can be simplified by finding the greatest common divisor (GCD) of 15 and 10, which is 5.
Here are the steps to simplify coefficients:

• Find the GCD of the numerator and the denominator.
• Divide both the numerator and the denominator by their GCD.

So, \(\frac{15}{10}\) simplifies to \(\frac{3}{2}\) because 15 divided by 5 equals 3, and 10 divided by 5 equals 2.
Simplifying coefficients reduces the overall complexity of the initial algebraic fraction and makes it much easier to solve or further simplify the given expression.

Exponent rules govern how we handle exponents when simplifying expressions. Here, we will focus on the rules relevant to our problem. The first rule is the product rule: \(a^m \times a^n = a^{m+n}\). This rule tells us if we are multiplying the same base, we add the exponents.
The second rule is the power rule: \((a^m)^n = a^{m \times n}\). This rule indicates that when raising a power to another power, we multiply the exponents.
Let's simplify the exponents in our given expression:

• For \(m\): Applying the product rule, \(m^{5} \times m^{-10} = m^{5-10} = m^{-5}\). Here, we subtracted the exponents.
• For \(n\): Again, using the product rule, \(n^{3} \times n^{-(-4)} = n^{3+4} = n^{7}\). Adding the exponents here because two negatives make a positive.

Knowing and correctly applying these rules helps in simplifying the expression systematically.

It's often helpful to rewrite any negative exponents as positive exponents to simplify the final expression. As previously explained, a negative exponent can be turned positive by taking the reciprocal of the base.
To illustrate: \(m^{-5} = \frac{1}{m^5}\). This step makes it easier to visualize and interpret the term within the fraction.
We rewrite the whole expression \(\frac{3n^7}{2m^5}\) so that all exponents are positive:

• \(\frac{3}{2}\) remains the same because it is already simplified.
• For positive exponents, \(n^7\) stays as it is.
• The term \(m^{-5}\) changes to \(\frac{1}{m^5}\) and gets moved to the denominator.

Combining these, we get \(\frac{3n^7}{2m^5}\).
This clearer form of expression using only positive exponents can be more easily understood and used in further mathematical operations.