To determine which expression is equivalent to [tex]\(\left(\frac{m^5 n}{p q^2}\right)^4\)[/tex], we will use the laws of exponents to simplify the expression step-by-step.
Starting with the given expression:
[tex]\[
\left(\frac{m^5 n}{p q^2}\right)^4
\][/tex]
Apply the power of a quotient rule, which states [tex]\(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\)[/tex]:
[tex]\[
\left(\frac{m^5 n}{p q^2}\right)^4 = \frac{(m^5 n)^4}{(p q^2)^4}
\][/tex]
First, simplify the numerator [tex]\((m^5 n)^4\)[/tex] by using the power rule [tex]\((a^m b^n)^p = a^{m \cdot p} b^{n \cdot p}\)[/tex]:
[tex]\[
(m^5 n)^4 = m^{5 \cdot 4} \cdot n^{1 \cdot 4} = m^{20} n^4
\][/tex]
Next, simplify the denominator [tex]\((p q^2)^4\)[/tex] by applying the same power rule:
[tex]\[
(p q^2)^4 = p^{1 \cdot 4} \cdot q^{2 \cdot 4} = p^4 q^8
\][/tex]
Combine the simplified numerator and denominator:
[tex]\[
\frac{m^{20} n^4}{p^4 q^8}
\][/tex]
Therefore, the simplified expression equivalent to [tex]\(\left(\frac{m^5 n}{p q^2}\right)^4\)[/tex] is:
[tex]\[
\frac{m^{20} n^4}{p^4 q^8}
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{\frac{m^{20} n^4}{p^4 q^8}}
\][/tex]
