- Divide the numerical coefficients: \(\frac{-24}{8} = -3\).
- Subtract the exponents for \(m\): \(m^{5 - (-7)} = m^{12}\).
- Subtract the exponents for \(n\): \(n^{4 - (-2)} = n^{6}\).
- Combine all parts to get the final simplified expression: \(\boxed{-3 m^{12} n^6}\).
Explanation
- Problem Analysis
The goal is to simplify the expression \(\frac{-24 m^5 n^4}{8 m^{-7} n^{-2}}\) given that \(m
eq 0\) and \(n
eq 0\). We can break this expression down by separating the coefficients, the powers of \(m\), and the powers of \(n\). The laws of exponents we will utilize are \(\frac{a^x}{a^y} = a^{x-y}\) and \(\frac{1}{a^{-y}} = a^y\).
- Dividing Coefficients
First, we divide the constant coefficients:
\[\frac{-24}{8} = -3\]
This gives us the numerical part of our simplified expression.
- Simplifying Powers of \(m\)
Next, we simplify the terms involving the variable \(m\). Using the division rule for exponents, \(\frac{m^5}{m^{-7}} = m^{5 - (-7)}\). Simplifying the exponent, we get:
\[m^{5+7} = m^{12}\]
Thus, the \(m\) component of our expression becomes \(m^{12}\).
- Simplifying Powers of \(n\)
Similarly, we simplify the terms involving the variable \(n\). Applying the division rule for exponents, \(\frac{n^4}{n^{-2}} = n^{4 - (-2)}\). Simplifying the exponent, we get:
\[n^{4+2} = n^6\]
Thus, the \(n\) component of our expression becomes \(n^6\).
- Final Result
Combining the results from the previous steps, we multiply the coefficient and the simplified variable powers together:
\[-3 \cdot m^{12} \cdot n^6 = -3 m^{12} n^6\]
This matches the third option provided. Therefore, the simplified form is \(\boxed{-3 m^{12} n^6}\).
Examples
In physics and engineering, simplified expressions are crucial for calculating physical quantities like work, power, or velocity when variables are raised to different powers. For instance, if a formula for energy density involves dividing terms with different exponents, simplifying them allows for easier calculations and better physical interpretation of the relationship between variables such as mass (\(m\)) and particle density (\(n\)). By mastering exponent rules, you can handle complex formulas in fields ranging from circuit design to fluid mechanics.