Find x, if `(3/4)^(5+2x)xx (3/4)^(11-x)= (3/4)^(8x-5)`

Answer

To solve the equation \((\frac{3}{4})^{5+2x} \cdot (\frac{3}{4})^{11-x} = (\frac{3}{4})^{8x-5}\), we can follow these steps:

### Step-by-Step Solution:

1. **Combine the Left Side Using the Property of Exponents**:
We know that \(a^m \cdot a^n = a^{m+n}\). So, we can combine the left side:
\[ (\frac{3}{4})^{(5 + 2x) + (11 - x)} = (\frac{3}{4})^{(8x - 5)} \]

2. **Simplify the Exponent on the Left Side**:
Now, simplify the exponent on the left side:
\[ 5 + 2x + 11 - x = 8x - 5 \]
This simplifies to:
\[ 16 + x = 8x - 5 \]

3. **Rearrange the Equation**:
Move all terms involving \(x\) to one side and constant terms to the other side:
\[ x - 8x = -5 - 16 \]
This leads to:
\[ -7x = -21 \]

4. **Solve for \(x\)**:
Divide both sides by -7:
\[ x = \frac{-21}{-7} = 3 \]

Thus, the solution is:
\[ \boxed{3} \]