- Expand the numerator: \((2mn)^4 = 16m^4n^4\).
- Simplify the numerical coefficient: \(\frac{16}{6} = \frac{8}{3}\).
- Use the quotient rule for exponents to simplify variables: \(m^4/m^{-3} = m^7\) and \(n^4/n^{-2} = n^6\).
- Combine to find the final simplified expression: \(\boxed{\frac{8m^7n^6}{3}}\).
Explanation
- Expanding the numerator
To simplify the expression \(\frac{(2mn)^4}{6m^{-3}n^{-2}}\), we first focus on the numerator. By applying the power of a product rule \((ab)^k = a^k b^k\), we have \((2mn)^4 = 2^4 \cdot m^4 \cdot n^4 = 16m^4n^4\). The given expression is now \(\frac{16m^4n^4}{6m^{-3}n^{-2}}\).
- Simplifying the coefficient
Next, we simplify the coefficient by dividing the constants in the numerator and denominator: \(\frac{16}{6} = \frac{8}{3}\). This gives us \(\frac{8}{3} \cdot \frac{m^4n^4}{m^{-3}n^{-2}}\).
- Applying quotient rule for exponents and final simplification
Now we apply the quotient rule for exponents \(\frac{a^x}{a^y} = a^{x-y}\) to the variables \(m\) and \(n\). For \(m\), we have \(\frac{m^4}{m^{-3}} = m^{4 - (-3)} = m^{4+3} = m^7\). For \(n\), we have \(\frac{n^4}{n^{-2}} = n^{4 - (-2)} = n^{4+2} = n^6\). Combining these results, we get the simplified expression \(\frac{8m^7n^6}{3}\).
Examples
Algebraic expressions like this are common in engineering and physics, where variables represent physical quantities such as mass (m) or amount of substance (n). Simplifying these expressions is essential for efficient calculation and problem-solving, much like how a carpenter simplifies a set of blueprints before cutting wood, ensuring the work is precise and saving time. Mastering these rules allows for the manipulation of complex formulas, making them easier to read and evaluate.