If `(a)/(b)=(c)/(d)=(e)/(f)`, then `(a^(3)+c^(3)+e^(3))/(b^(3)+d^(3)+f^(3))=_____`

Options

A. 1
B. `(a+c+e)/(b+d+f)`
C. `((a+c+e)^(2))/((b+d+f)^(2))`
D. `(ace)/(bdf)`

Explanation

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

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

### Step 1: Express \(a\), \(c\), and \(e\) in terms of \(k\)
From the above relationship, we can express \(a\), \(c\), and \(e\) as follows:

\[ a = bk, \quad c = dk, \quad e = fk \]

### Step 2: Calculate \(a^3\), \(c^3\), and \(e^3\)
Now we will cube each of these expressions:

\[ a^3 = (bk)^3 = b^3k^3 \]
\[ c^3 = (dk)^3 = d^3k^3 \]
\[ e^3 = (fk)^3 = f^3k^3 \]

### Step 3: Find \(a^3 + c^3 + e^3\)
Now, we can add these cubes together:

\[ a^3 + c^3 + e^3 = b^3k^3 + d^3k^3 + f^3k^3 \]
\[ = k^3(b^3 + d^3 + f^3) \]

### Step 4: Calculate \(b^3\), \(d^3\), and \(f^3\)
Next, we need to find the denominator \(b^3 + d^3 + f^3\), which we already have:

\[ b^3 + d^3 + f^3 \]

### Step 5: Form the final expression
Now we can form the final expression:

\[ \frac{a^3 + c^3 + e^3}{b^3 + d^3 + f^3} = \frac{k^3(b^3 + d^3 + f^3)}{b^3 + d^3 + f^3} \]

### Step 6: Simplify the expression
Since \(b^3 + d^3 + f^3\) is common in both the numerator and the denominator, we can simplify:

\[ = k^3 \]

### Step 7: Substitute back \(k\)
Recalling that \(k = \frac{a}{b} = \frac{c}{d} = \frac{e}{f}\), we can express \(k^3\) as:

\[ k^3 = \left(\frac{a}{b}\right)^3 = \frac{a^3}{b^3} \]

Thus, we conclude:

\[ \frac{a^3 + c^3 + e^3}{b^3 + d^3 + f^3} = \frac{a^3}{b^3} \]

### Final Answer
The final answer is:

\[ \frac{a^3 + c^3 + e^3}{b^3 + d^3 + f^3} = \frac{a^3}{b^3} \]