To simplify the expression [tex]\(\left(\frac{3m}{n}\right)^4\)[/tex], we need to apply the power to both the numerator and the denominator inside the parentheses.
1. Simplifying the Numerator: The numerator of the expression is [tex]\(3m\)[/tex]. When we raise it to the power of 4, we apply the exponent to both the constant and the variable separately:
[tex]\[
(3m)^4 = 3^4 \cdot m^4
\][/tex]
We know that [tex]\(3^4 = 81\)[/tex], so this becomes:
[tex]\[
3^4 \cdot m^4 = 81m^4
\][/tex]
2. Simplifying the Denominator: The denominator of the expression is [tex]\(n\)[/tex]. Raising [tex]\(n\)[/tex] to the power of 4 gives:
[tex]\[
n^4
\][/tex]
3. Combining the Results: After simplifying both the numerator and the denominator, we combine them to get:
[tex]\[
\frac{(3m)^4}{n^4} = \frac{81m^4}{n^4}
\][/tex]
So, the correct simplification of the expression [tex]\(\left(\frac{3m}{n}\right)^4\)[/tex] is:
[tex]\[
\frac{81m^4}{n^4}
\][/tex]
Therefore, the correct answer among the given choices is:
[tex]\[
\boxed{\frac{81m^4}{n^4}}
\][/tex]
