If ` ((5)/(12)) ^(-4) xx ((5)/(12)) ^(3x) = ((5)/(12))^(5),` then `x = ` ?

Options

A. `-1`
B. 1
C. 2
D. 3

Answer

To solve the equation \(\left(\frac{5}{12}\right)^{-4} \times \left(\frac{5}{12}\right)^{3x} = \left(\frac{5}{12}\right)^{5}\), we will use the properties of exponents.

### Step-by-step Solution:

1. **Identify the bases and apply the property of exponents**:
Since both sides of the equation have the same base \(\frac{5}{12}\), we can use the property of exponents that states \(a^m \times a^n = a^{m+n}\).
\[ \left(\frac{5}{12}\right)^{-4} \times \left(\frac{5}{12}\right)^{3x} = \left(\frac{5}{12}\right)^{-4 + 3x} \]

2. **Set the exponents equal to each other**:
Now we can equate the exponents since the bases are the same:
\[ -4 + 3x = 5 \]

3. **Solve for \(x\)**:
To isolate \(3x\), add \(4\) to both sides of the equation:
\[ 3x = 5 + 4 \]
\[ 3x = 9 \]

4. **Divide by \(3\)**:
Now, divide both sides by \(3\) to solve for \(x\):
\[ x = \frac{9}{3} \]
\[ x = 3 \]

### Final Answer:
Thus, the value of \(x\) is \(3\).