Find 'a' such that `((6)/(7))^(a)xx((6)/(7))^(3a)=(1296)/(2401)`.

Options

A. `4`
B. `1`
C. `0`
D. `5`

Answer

To solve the equation \(\left(\frac{6}{7}\right)^{a} \times \left(\frac{6}{7}\right)^{3a} = \frac{1296}{2401}\), we can follow these steps:

### Step 1: Combine the exponents on the left side
Using the property of exponents that states \(a^m \times a^n = a^{m+n}\), we can combine the left side:

\[ \left(\frac{6}{7}\right)^{a + 3a} = \left(\frac{6}{7}\right)^{4a} \]

### Step 2: Rewrite the right side
Next, we need to express the right side \(\frac{1296}{2401}\) in terms of powers of \(\frac{6}{7}\).

First, we can factor \(1296\) and \(2401\):
- \(1296 = 6^4\) (since \(6 \times 6 \times 6 \times 6 = 1296\))
- \(2401 = 7^4\) (since \(7 \times 7 \times 7 \times 7 = 2401\))

Thus, we can rewrite the right side:

\[ \frac{1296}{2401} = \frac{6^4}{7^4} = \left(\frac{6}{7}\right)^4 \]

### Step 3: Set the exponents equal
Now we have:

\[ \left(\frac{6}{7}\right)^{4a} = \left(\frac{6}{7}\right)^{4} \]

Since the bases are the same, we can set the exponents equal to each other:

\[ 4a = 4 \]

### Step 4: Solve for \(a\)
Now, we can solve for \(a\):

\[ a = \frac{4}{4} = 1 \]

### Final Answer
Thus, the value of \(a\) is:

\[ \boxed{1} \]