- Simplify the quotient inside the parentheses: \(\frac{4mn}{m^{-2}n^6} = 4m^3n^{-5}\).
- Apply the outer exponent \(-2\) to each factor in the term: \((4m^3n^{-5})^{-2} = 4^{-2} \cdot (m^3)^{-2} \cdot (n^{-5})^{-2}\).
- Calculate the values: \(\frac{1}{16} \cdot m^{-6} \cdot n^{10}\).
- Rewrite the final expression as \(\boxed{\frac{n^{10}}{16 m^6}}\).
Explanation
- Problem Analysis and Properties of Exponents
We need to simplify the algebraic expression \(\left(\frac{4 m n}{m^{-2} n^6}\right)^{-2}\) under the condition that \(m
eq 0\) and \(n
eq 0\). We will apply the properties of exponents to reach the simplest form, which will allow us to identify the correct equivalent expression from the given options. The relevant rules are: \((x^a)^b = x^{ab}\), \(\frac{x^a}{x^b} = x^{a-b}\), \(x^{-a} = \frac{1}{x^a}\), and \((xyz)^a = x^a y^a z^a\).
- Simplifying the Interior Expression
First, let us simplify the expression inside the parentheses: \(\frac{4 m n}{m^{-2} n^6}\). Using the quotient rule \(\frac{x^a}{x^b} = x^{a-b}\), we handle the variables \(m\) and \(n\) separately:
- For \(m\): \(\frac{m^1}{m^{-2}} = m^{1 - (-2)} = m^{1+2} = m^3\).
- For \(n\): \(\frac{n^1}{n^6} = n^{1 - 6} = n^{-5}\).
Thus, the expression inside the parentheses becomes \(4 m^3 n^{-5}\).
- Applying the Outer Exponent and Final Simplification
Now, we apply the outer exponent of \(-2\) to our simplified interior expression: \((4 m^3 n^{-5})^{-2}\).
Using the power of a product rule \((xyz)^a = x^a y^a z^a\), we distribute the \(-2\) exponent to each factor:
\((4 m^3 n^{-5})^{-2} = 4^{-2} \cdot (m^3)^{-2} \cdot (n^{-5})^{-2}\).
Calculating each part:
- \(4^{-2} = \frac{1}{4^2} = \frac{1}{16}\).
- \((m^3)^{-2} = m^{3 \times (-2)} = m^{-6} = \frac{1}{m^6}\).
- \((n^{-5})^{-2} = n^{-5 \times (-2)} = n^{10}\).
Combining these, we get \(\frac{1}{16} \cdot \frac{1}{m^6} \cdot n^{10} = \frac{n^{10}}{16 m^6}\).
- Comparison and Conclusion
Comparing our result \(\frac{n^{10}}{16 m^6}\) with the provided options:
- \(\frac{n^6}{16 m^8}\)
- \(\frac{n^{10}}{16 m^6}\)
- \(\frac{n^{10}}{8 m^8}\)
- \(\frac{4 m^3}{n^8}\)
The simplified expression matches the second option. Thus, \(\left(\frac{4 m n}{m^{-2} n^6}\right)^{-2} = \boxed{\frac{n^{10}}{16 m^6}}\).
Examples
Understanding exponent rules is crucial in scientific fields, such as chemistry or physics, where very large or very small numbers are often represented in scientific notation. For example, if a chemist is calculating the concentration of a substance and encounters a term with negative exponents resulting from inverse measurements, they can use these simplification rules to rewrite the expression into a more manageable form, like \(\frac{n^{10}}{16 m^6}\), which is much easier to compute and interpret during data analysis.