- Apply exponent properties \((abc)^k = a^k b^k c^k\) and \((a^b)^c = a^{bc}\) to expand both the numerator and denominator.
- Use the rule \(\frac{x^a}{x^b} = x^{a-b}\) to simplify the variables \(m\) and \(n\).
- Simplify the numerical coefficients \(3^4=81\) and \(2^3=8\).
- The simplified expression is \(\boxed{\frac{81 m^2 n^5}{8}}\).
Explanation
- Analyzing the Expression
We are given the expression \(\frac{\left(3 m^{-1} n^2\right)^4}{\left(2 m^{-2} n\right)^3}\) with the constraints \(m
eq 0\) and \(n
eq 0\). Our goal is to simplify this expression using the standard properties of exponents: \((ab)^k = a^k b^k\), \((a^b)^c = a^{bc}\), \(\frac{a^m}{a^n} = a^{m-n}\), and \(a^{-n} = \frac{1}{a^n}\). Analyzing the numerator and the denominator separately will allow us to break down the problem efficiently.
- Applying Rules of Exponents
First, simplify the numerator \(\left(3 m^{-1} n^2\right)^4\) by applying the power of a product rule \((abc)^k = a^k b^k c^k\):
\[\left(3 m^{-1} n^2\right)^4 = 3^4 \cdot (m^{-1})^4 \cdot (n^2)^4 = 81 \cdot m^{-4} \cdot n^8.\]
Next, simplify the denominator \(\left(2 m^{-2} n\right)^3\) using the same rule:
\[\left(2 m^{-2} n\right)^3 = 2^3 \cdot (m^{-2})^3 \cdot n^3 = 8 \cdot m^{-6} \cdot n^3.\]
Now combine them into the fraction:
\[\frac{81 \cdot m^{-4} \cdot n^8}{8 \cdot m^{-6} \cdot n^3} = \frac{81}{8} \cdot \frac{m^{-4}}{m^{-6}} \cdot \frac{n^8}{n^3}.\]
- Simplifying and Finalizing the Expression
Now simplify the powers of \(m\) and \(n\) using the division rule \(\frac{x^a}{x^b} = x^{a-b}\):
For \(m\): \(\frac{m^{-4}}{m^{-6}} = m^{-4 - (-6)} = m^{-4+6} = m^2\).
For \(n\): \(\frac{n^8}{n^3} = n^{8-3} = n^5\).
Putting it all together, the expression simplifies to \(\frac{81}{8} m^2 n^5\). Note that the options provided seem to contain a typo for the exponent of \(n\), but based on the algebraic rules applied, the result is \(\frac{81}{8} m^2 n^5\). Given the options provided, \(\frac{81 m^2 n^5}{8}\) is the logically derived equivalent.
Examples
In computer science and digital data storage, simplifying exponential expressions is fundamental when calculating file sizes, memory addressing, or data transmission rates. For example, if a data compression algorithm processes packets of size \((3 m^{-1} n^2)^4\) and divides them into groups of \((2 m^{-2} n)^3\), simplifying the expression helps programmers optimize memory allocation and improve system performance. Mathematical mastery of exponent laws is crucial for creating efficient software and hardware designs.