Let's simplify the given expression [tex]\(\left(\frac{m^5 n}{\rho q^2}\right)^4\)[/tex] step by step.
1. Original Expression:
[tex]\[\left(\frac{m^5 n}{\rho q^2}\right)^4\][/tex]
2. Apply the exponent to both numerator and denominator separately:
[tex]\[(\frac{m^5 n}{\rho q^2})^4 = \frac{(m^5 n)^4}{(\rho q^2)^4}\][/tex]
3. Simplify the numerator:
[tex]\[(m^5 n)^4 = m^{5 \cdot 4} \cdot n^4 = m^{20} \cdot n^4\][/tex]
4. Simplify the denominator:
[tex]\[(\rho q^2)^4 = \rho^4 \cdot (q^2)^4 = \rho^4 \cdot q^{2 \cdot 4} = \rho^4 \cdot q^8\][/tex]
5. Combine the simplified numerator and denominator:
[tex]\[\frac{m^{20} \cdot n^4}{\rho^4 \cdot q^8}\][/tex]
Therefore, the simplified and equivalent expression is:
[tex]\[\frac{m^{20} n^4}{\rho^4 q^8}\][/tex]
Comparing this with the given options:
- [tex]\(\frac{m^9 n^5}{p^5 q^6}\)[/tex]
- [tex]\(\frac{m^{20} n^4}{p q^2}\)[/tex]
- [tex]\(\frac{m^{20} n^4}{p^4 q^8}\)[/tex]
- [tex]\(\frac{m^9 n^4}{p^4 q^6}\)[/tex]
The correct expression that matches is:
[tex]\[\boxed{\frac{m^{20} n^4}{p^4 q^8}}\][/tex]
