If `(a)/(b)=(3)/(4),(b)/(c)=(4)/(5)and(c)/(d)=(5)/(6)`, then the sum of the numerator and the denominator (which are coprimes) of `((a)/(d))^(10)` is.

Options

A. 1025
B. 4097
C. 2049
D. 513

Answer

To solve the problem, we need to find the value of \((a/d)^{10}\) given the ratios \((a/b) = (3/4)\), \((b/c) = (4/5)\), and \((c/d) = (5/6)\). We will follow these steps:

### Step 1: Express \(a\), \(b\), \(c\), and \(d\) in terms of a common variable.

From the given ratios, we can express \(a\), \(b\), \(c\), and \(d\) in terms of \(d\):

1. From \((c/d) = (5/6)\):
\[ c = \frac{5}{6}d \]

2. From \((b/c) = (4/5)\):
\[ b = \frac{4}{5}c = \frac{4}{5} \cdot \frac{5}{6}d = \frac{4}{6}d = \frac{2}{3}d \]

3. From \((a/b) = (3/4)\):
\[ a = \frac{3}{4}b = \frac{3}{4} \cdot \frac{2}{3}d = \frac{2}{4}d = \frac{1}{2}d \]

### Step 2: Find \(\frac{a}{d}\).

Now that we have \(a\) in terms of \(d\):
\[ \frac{a}{d} = \frac{\frac{1}{2}d}{d} = \frac{1}{2} \]

### Step 3: Calculate \(\left(\frac{a}{d}\right)^{10}\).

Now we raise \(\frac{a}{d}\) to the power of 10:
\[ \left(\frac{a}{d}\right)^{10} = \left(\frac{1}{2}\right)^{10} = \frac{1^{10}}{2^{10}} = \frac{1}{1024} \]

### Step 4: Identify the numerator and denominator.

The fraction \(\frac{1}{1024}\) has a numerator of 1 and a denominator of 1024. Since 1 and 1024 are coprime (the only common divisor is 1), we can proceed to find their sum.

### Step 5: Calculate the sum of the numerator and denominator.

The sum of the numerator and denominator is:
\[ 1 + 1024 = 1025 \]

### Final Answer:
The sum of the numerator and the denominator (which are coprimes) of \(\left(\frac{a}{d}\right)^{10}\) is **1025**.

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