Question

Consider the expression.

(StartFraction 2 m Superscript negative 1 baseline n Superscript 5 Baseline Over 3 m Superscript 0 Baseline n Superscript 4 Baseline EndFraction) squared

What is the value of the expression if m = –5 and n = 3?
Negative StartFraction 24 Over 25 EndFraction
Negative StartFraction 4 Over 25 EndFraction
StartFraction 4 Over 25 EndFraction
StartFraction 24 Over 25 EndFraction

Answer 1

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:

\(\frac{4}{25}\)