Question

which expression is equivalent to \\(\frac{(2mn)^4}{6m^{-3}n^{-2}}\\)? assume \\(m \
eq 0, n \
eq 0\\). \\(\frac{m^4n^6}{3}\\) \\(\frac{8m^7n^6}{3}\\) \\(\frac{8m^{10}n^{12}}{3}\\) \\(\frac{10m^7n^6}{3}\\)

Explanation:

Step1: Expand the numerator

Using the power of a product rule \((ab)^n = a^n b^n\), we expand \((2mn)^4\) as \(2^4m^4n^4\). So \(2^4 = 16\), thus the numerator becomes \(16m^4n^4\).

Step2: Simplify the fraction

We have the fraction \(\frac{16m^4n^4}{6m^{-4}n^{-2}}\). First, simplify the coefficients: \(\frac{16}{6}=\frac{8}{3}\). Then, for the variables with exponents, use the rule \(\frac{a^m}{a^n}=a^{m - n}\). For \(m\): \(m^{4-(-4)} = m^{4 + 4}=m^8\)? Wait, no, wait, wait. Wait, the denominator is \(m^{-4}\), so \(\frac{m^4}{m^{-4}}=m^{4-(-4)}=m^{8}\)? Wait, no, the original problem's denominator is \(6m^{-4}n^{-2}\), numerator is \(16m^4n^4\). So for \(m\): \(m^{4-(-4)} = m^{8}\)? Wait, no, wait, maybe I misread the problem. Wait, the problem is \(\frac{(2mn)^4}{6m^{-4}n^{-2}}\)? Wait, no, looking back, the user's problem: "Which expression is equivalent to \(\frac{(2mn)^4}{6m^{-4}n^{-2}}\)? Assume \(m
eq0,n
eq0\)." Wait, no, maybe the denominator is \(6m^{-4}n^{-2}\)? Wait, no, the original image: maybe the denominator is \(6m^{-4}n^{-2}\)? Wait, no, let's re-express.

Wait, \((2mn)^4 = 2^4m^4n^4 = 16m^4n^4\). Then the denominator is \(6m^{-4}n^{-2}\). So when we divide, we subtract exponents: for \(m\): \(m^{4-(-4)} = m^{8}\), for \(n\): \(n^{4-(-2)} = n^{6}\), and the coefficient is \(16/6 = 8/3\). Wait, but the options have \(m^7\)? Wait, maybe I misread the denominator. Wait, maybe the denominator is \(6m^{-3}n^{-2}\)? No, the user's problem: let's check again. Wait, the options have \(\frac{8m^7n^6}{3}\)? Wait, no, maybe the denominator is \(6m^{-3}n^{-2}\)? Wait, no, the original problem: "Which expression is equivalent to \(\frac{(2mn)^4}{6m^{-4}n^{-2}}\)? Assume \(m
eq0,n
eq0\)." Wait, no, maybe a typo? Wait, no, let's recalculate.

Wait, \((2mn)^4 = 16m^4n^4\). Denominator: \(6m^{-4}n^{-2}\). So \(\frac{16m^4n^4}{6m^{-4}n^{-2}}=\frac{16}{6}m^{4 - (-4)}n^{4 - (-2)}=\frac{8}{3}m^{8}n^{6}\)? But that's not in the options. Wait, maybe the denominator is \(6m^{-3}n^{-2}\)? Wait, no, the options have \(\frac{8m^7n^6}{3}\), \(\frac{8m^{10}n^{12}}{3}\), etc. Wait, maybe I misread the numerator or denominator. Wait, maybe the original problem is \(\frac{(2mn)^4}{6m^{-3}n^{-2}}\)? No, let's check the options. The second option is \(\frac{8m^7n^6}{3}\), third is \(\frac{8m^{10}n^{12}}{3}\), first is \(\frac{m^4n^6}{3}\), fourth is \(\frac{10m^7n^6}{3}\). Wait, maybe the numerator is \((2m^3n)^4\)? No, the user wrote \((2mn)^4\). Wait, maybe the denominator is \(6m^{-3}n^{-2}\)? Wait, no, let's re-express:

Wait, \((2mn)^4 = 16m^4n^4\). Denominator: \(6m^{-4}n^{-2}\). So \(m^{4 - (-4)} = m^8\), \(n^{4 - (-2)} = n^6\), coefficient \(16/6 = 8/3\). But that's not in the options. Wait, maybe the denominator is \(6m^{-3}n^{-2}\)? Then \(m^{4 - (-3)} = m^7\), \(n^{4 - (-2)} = n^6\), coefficient \(16/6 = 8/3\). Ah! Maybe the denominator is \(6m^{-3}n^{-2}\), not \(m^{-4}\). Maybe a typo in the problem, or maybe I misread. Let's assume that the denominator is \(6m^{-3}n^{-2}\). Then:

Step1: Expand numerator

\((2mn)^4 = 16m^4n^4\)

Step2: Simplify fraction

Coefficient: \(16/6 = 8/3\)
For \(m\): \(m^{4 - (-3)} = m^{4 + 3} = m^7\)
For \(n\): \(n^{4 - (-2)} = n^{4 + 2} = n^6\)
Thus, the expression becomes \(\frac{8m^7n^6}{3}\), which is the second option.

Answer:

\(\frac{8m^7n^6}{3}\) (the second option, e.g., B. \(\frac{8m^7n^6}{3}\))