Let's simplify the given expression step by step:

$\left(\frac{3m^{-5}n^2}{4m^{-2}n^0}\right)^2\cdot\left(\frac{mn^4}{9n}\right)^2$

Step 1: Simplify Each Fraction Inside the Parentheses

First Fraction:

$\frac{3m^{-5}n^2}{4m^{-2}n^0}$

• Since $n^0 = 1$, we ignore it.
• Divide the powers of $m$:

$m^{-5} \div m^{-2} = m^{-5 - (-2)} = m^{-3}$

Thus, the first fraction simplifies to:

$\frac{3m^{-3}n^2}{4}$

Second Fraction:

$\frac{mn^4}{9n}$

• Divide the powers of $n$:

$n^4 \div n = n^{4-1} = n^3$

Thus, the second fraction simplifies to:

$\frac{mn^3}{9}$

Step 2: Square Each Simplified Fraction

First Squared Expression:

$\left(\frac{3m^{-3}n^2}{4}\right)^2$

Squaring each term:

$\frac{3^2 m^{-6} n^4}{4^2} = \frac{9m^{-6}n^4}{16}$

Second Squared Expression:

$\left(\frac{mn^3}{9}\right)^2$

Squaring each term:

$\frac{m^2 n^6}{81}$

Step 3: Multiply the Two Results

$\left(\frac{9m^{-6}n^4}{16}\right) \cdot \left(\frac{m^2 n^6}{81}\right)$

Multiply the numerators:

$9m^{-6}n^4 \times m^2 n^6 = 9m^{-6+2}n^{4+6} = 9m^{-4}n^{10}$

Multiply the denominators:

$16 \times 81 = 1296$

Thus, the expression simplifies to:

$\frac{9m^{-4}n^{10}}{1296}$

Step 4: Simplify the Coefficient

$\frac{9}{1296} = \frac{1}{144}$

Thus, the final simplified expression is:

$\frac{m^{-4}n^{10}}{144}$

Or, writing with positive exponents:

$\frac{n^{10}}{144m^4}$

Final Answer:

$\frac{n^{10}}{144m^4}$

Would you like me to clarify any step or go into more details?

Related Questions:

1. How do you simplify expressions with negative exponents?
2. What are the laws of exponents and how do they apply to fraction simplifications?
3. How do you handle complex fractions with variables in exponent form?
4. What is the process of rationalizing expressions with negative exponents?
5. How can exponent rules be applied in algebraic fraction simplifications?

Tip: Always simplify inside parentheses before applying exponent rules!