If ` ((3(1)/(4) )^(4) -(4( 1)/(3))^(4))/( (3,(1)/(4) )^(2) -(4,(1)/(3))^(2) )= ((13a)/(12))^(2) ,` find a.

Answer

To solve the given equation

\[ \frac{(3 \cdot \frac{1}{4})^4 - (4 \cdot \frac{1}{3})^4}{(3 \cdot \frac{1}{4})^2 - (4 \cdot \frac{1}{3})^2} = \left(\frac{13a}{12}\right)^2, \]

we will follow these steps:

### Step 1: Simplify the numerator and denominator
First, we simplify the terms in the numerator and denominator.

- The numerator is:
\[ (3 \cdot \frac{1}{4})^4 - (4 \cdot \frac{1}{3})^4 = \left(\frac{3}{4}\right)^4 - \left(\frac{4}{3}\right)^4. \]

- The denominator is:
\[ (3 \cdot \frac{1}{4})^2 - (4 \cdot \frac{1}{3})^2 = \left(\frac{3}{4}\right)^2 - \left(\frac{4}{3}\right)^2. \]

### Step 2: Apply the difference of squares
We can use the difference of squares formula, \( a^2 - b^2 = (a - b)(a + b) \).

Let:
- \( a = \frac{3}{4} \)
- \( b = \frac{4}{3} \)

Thus, we have:
\[ \left(\frac{3}{4}\right)^4 - \left(\frac{4}{3}\right)^4 = \left(\frac{3}{4} - \frac{4}{3}\right)\left(\frac{3}{4} + \frac{4}{3}\right). \]

### Step 3: Calculate \( a - b \) and \( a + b \)
Calculating \( a - b \):
\[ \frac{3}{4} - \frac{4}{3} = \frac{9 - 16}{12} = -\frac{7}{12}. \]

Calculating \( a + b \):
\[ \frac{3}{4} + \frac{4}{3} = \frac{9 + 16}{12} = \frac{25}{12}. \]

### Step 4: Substitute back into the numerator
The numerator becomes:
\[ \left(-\frac{7}{12}\right)\left(\frac{25}{12}\right) = -\frac{175}{144}. \]

### Step 5: Simplify the denominator
The denominator is:
\[ \left(\frac{3}{4}\right)^2 - \left(\frac{4}{3}\right)^2 = \left(\frac{3}{4} - \frac{4}{3}\right)\left(\frac{3}{4} + \frac{4}{3}\right). \]
Using the same calculations for \( a - b \) and \( a + b \):
\[ \left(\frac{3}{4}\right)^2 - \left(\frac{4}{3}\right)^2 = \left(-\frac{7}{12}\right)\left(\frac{25}{12}\right) = -\frac{175}{144}. \]

### Step 6: Substitute into the equation
Now substituting back into the equation:
\[ \frac{-\frac{175}{144}}{-\frac{175}{144}} = 1. \]

Thus, we have:
\[ 1 = \left(\frac{13a}{12}\right)^2. \]

### Step 7: Solve for \( a \)
Taking square roots:
\[ \frac{13a}{12} = 1 \quad \text{or} \quad \frac{13a}{12} = -1. \]

From \( \frac{13a}{12} = 1 \):
\[ 13a = 12 \implies a = \frac{12}{13}. \]

From \( \frac{13a}{12} = -1 \):
\[ 13a = -12 \implies a = -\frac{12}{13}. \]

### Final Answer
Thus, the values of \( a \) are:
\[ a = \frac{12}{13} \quad \text{or} \quad a = -\frac{12}{13}. \]