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 given problem, we start with the equation:

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

From this, we can express \(a\), \(c\), and \(e\) in terms of \(k\), \(b\), \(d\), and \(f\):

1. **Express \(a\), \(c\), and \(e\)**:
- \(a = 2bk\)
- \(c = dk\)
- \(e = 3fk\)

Now, we need to substitute these values into the expression we want to evaluate:

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

2. **Substituting \(a\), \(c\), and \(e\)** into the numerator:
- \(a^4 = (2bk)^4 = 16b^4k^4\)
- \(a^2 = (2bk)^2 = 4b^2k^2\)
- \(c^2 = (dk)^2 = d^2k^2\)
- \(e^4 = (3fk)^4 = 81f^4k^4\)

Now substituting these into the numerator:

\[ 2(16b^4k^4) b^2 + 3(4b^2k^2)(d^2k^2) - 5(81f^4k^4)f \]

This simplifies to:

\[ 32b^6k^4 + 12b^2d^2k^4 - 405f^5k^4 \]

3. **Factor out \(k^4\)** from the numerator:

\[ k^4(32b^6 + 12b^2d^2 - 405f^5) \]

4. **Now, for the denominator**, we have:

\[ 32b^6 + 12b^2d^2 - 405f^5 \]

5. **Putting it all together**, we have:

\[ \frac{k^4(32b^6 + 12b^2d^2 - 405f^5)}{32b^6 + 12b^2d^2 - 405f^5} \]

6. **Simplifying the expression**, we find that:

\[ = k^4 \]

7. **Finding the value of \(k\)**:
Since \(k\) is a common ratio, we can choose any convenient value for \(k\). For simplicity, let's take \(k = 1\):

Thus, \(k^4 = 1^4 = 1\).

**Final Answer**:
\[ \text{The value of the expression is } 1. \]