To evaluate the expression
\[ \left( \frac{2m^{-1} n^{5}}{3m^{0} n^{4}} \right)^{2} \]
we first simplify the inner fraction, and then we will substitute \(m = -5\) and \(n = 3\).
Step 1: Simplify the expression
The expression inside the square is:
\[ \frac{2m^{-1} n^{5}}{3m^{0} n^{4}} \]
We know that \(m^{0} = 1\), so:
\[ = \frac{2m^{-1} n^{5}}{3 \cdot 1 \cdot n^{4}} = \frac{2m^{-1} n^{5}}{3n^{4}} \]
Now we can simplify \(n^{5} / n^{4}\):
\[ = \frac{2m^{-1} n^{1}}{3} = \frac{2n}{3m} \]
Step 2: Square the expression
Now we square this simplified expression:
\[ \left( \frac{2n}{3m} \right)^{2} = \frac{(2n)^{2}}{(3m)^{2}} = \frac{4n^{2}}{9m^{2}} \]
Step 3: Substitute \(m = -5\) and \(n = 3\)
Substituting \(n = 3\) and \(m = -5\):
\[ = \frac{4(3)^{2}}{9(-5)^{2}} = \frac{4 \cdot 9}{9 \cdot 25} \]
Simplifying further:
\[ = \frac{36}{225} \]
Now we can simplify \(\frac{36}{225}\):
- The greatest common divisor (GCD) of 36 and 225 is 9:
- \(\frac{36}{9} = 4\)
- \(\frac{225}{9} = 25\)
Thus:
\[ = \frac{4}{25} \]
Final Answer
So, the value of the expression is
\[ \frac{4}{25} \]
The correct answer is:
\(\frac{4}{25}\)