If `(a)/(2b) = (c )/(d) = (e )/(3f)`, then the value of
`(2a^(4)b^(2)+3a^(2)c^(2)-5e^(4)f)/(32b^(6)+12b^(2)d^(2)-405f^(5))` is

Options

A. `(a^(4))/(16b^(4))`
B. `(16c^(4))/(d^(4))`
C. `(e^(4))/(27f^(4))`
D. `(81c^(4))/(d^(4))`

Explanation

To solve the problem, we start with the given equality:

\[ \frac{a}{2b} = \frac{c}{d} = \frac{e}{3f} = x \]

From this, we can express \(a\), \(c\), and \(e\) in terms of \(x\):

1. \(a = 2bx\)
2. \(c = dx\)
3. \(e = 3fx\)

Now, we will substitute these expressions into the numerator and denominator of the given expression:

\[ \frac{2a^4b^2 + 3a^2c^2 - 5e^4f}{32b^6 + 12b^2d^2 - 405f^5} \]

### Step 1: Substitute \(a\), \(c\), and \(e\) into the numerator

Substituting \(a\), \(c\), and \(e\):

- For \(2a^4b^2\):
\[ 2a^4b^2 = 2(2bx)^4b^2 = 2 \cdot 16b^4x^4 \cdot b^2 = 32b^6x^4 \]

- For \(3a^2c^2\):
\[ 3a^2c^2 = 3(2bx)^2(dx)^2 = 3 \cdot 4b^2x^2 \cdot d^2x^2 = 12b^2d^2x^4 \]

- For \(-5e^4f\):
\[ -5e^4f = -5(3fx)^4f = -5 \cdot 81f^4x^4 \cdot f = -405f^5x^4 \]

Combining these results, the numerator becomes:
\[ 32b^6x^4 + 12b^2d^2x^4 - 405f^5x^4 = x^4(32b^6 + 12b^2d^2 - 405f^5) \]

### Step 2: Substitute into the denominator

The denominator is:
\[ 32b^6 + 12b^2d^2 - 405f^5 \]

### Step 3: Combine the results

Now we can write the entire expression:
\[ \frac{x^4(32b^6 + 12b^2d^2 - 405f^5)}{32b^6 + 12b^2d^2 - 405f^5} \]

Since the numerator and denominator are the same (as long as \(32b^6 + 12b^2d^2 - 405f^5 \neq 0\)), we can simplify this to:
\[ x^4 \]

### Step 4: Express \(x^4\) in terms of \(a\), \(b\), \(c\), \(d\), \(e\), and \(f\)

Recall that \(x = \frac{a}{2b}\). Thus:
\[ x^4 = \left(\frac{a}{2b}\right)^4 = \frac{a^4}{16b^4} \]

### Final Result

The value of the expression is:
\[ \frac{a^4}{16b^4} \]