To determine which expression is equivalent to [tex]\(\left(\frac{m^5 n}{p q^2}\right)^4\)[/tex], we will simplify the given expression step-by-step.
Given expression: [tex]\(\left(\frac{m^5 n}{p q^2}\right)^4\)[/tex]
First, use the property of exponents that states [tex]\(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\)[/tex].
So, [tex]\(\left(\frac{m^5 n}{p q^2}\right)^4 = \frac{(m^5 n)^4}{(p q^2)^4}\)[/tex].
Next, raise each term inside the numerator and denominator to the power of 4:
Numerator:
[tex]\[
(m^5 n)^4 = (m^5)^4 \cdot (n)^4
= m^{5 \cdot 4} \cdot n^4
= m^{20} \cdot n^4
\][/tex]
Denominator:
[tex]\[
(p q^2)^4 = (p)^4 \cdot (q^2)^4
= p^4 \cdot (q^2)^4
= p^4 \cdot q^{2 \cdot 4}
= p^4 \cdot q^8
\][/tex]
Putting it all together, we get:
[tex]\[
\frac{(m^5 n)^4}{(p q^2)^4} = \frac{m^{20} \cdot n^4}{p^4 \cdot q^8}
\][/tex]
Thus, the 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]
Therefore, the correct option is:
[tex]\[
\boxed{\frac{m^{20} n^4}{p^4 q^8}}
\][/tex]
