Given ` ((8)/(27))^(x- 1) = ((9)/(4))^(2x+ 1)` , find the value of x .

Answer

To solve the equation \(\left(\frac{8}{27}\right)^{x-1} = \left(\frac{9}{4}\right)^{2x+1}\), we can follow these steps:

### Step 1: Rewrite the bases in terms of powers
We know that:
- \(8 = 2^3\)
- \(27 = 3^3\)
- \(9 = 3^2\)
- \(4 = 2^2\)

Thus, we can rewrite the equation as:
\[ \left(\frac{2^3}{3^3}\right)^{x-1} = \left(\frac{3^2}{2^2}\right)^{2x+1} \]

### Step 2: Simplify the fractions
This can be rewritten as:
\[ \left(\frac{2}{3}\right)^{3(x-1)} = \left(\frac{3}{2}\right)^{2(2x+1)} \]

### Step 3: Rewrite the right-hand side
Notice that \(\frac{3}{2} = \left(\frac{2}{3}\right)^{-1}\). Therefore, we can rewrite the right-hand side:
\[ \left(\frac{2}{3}\right)^{3(x-1)} = \left(\frac{2}{3}\right)^{-2(2x+1)} \]

### Step 4: Set the exponents equal
Since the bases are the same, we can equate the exponents:
\[ 3(x-1) = -2(2x+1) \]

### Step 5: Expand both sides
Expanding both sides gives:
\[ 3x - 3 = -4x - 2 \]

### Step 6: Combine like terms
Now, we can add \(4x\) to both sides and add \(3\) to both sides:
\[ 3x + 4x = -2 + 3 \]
\[ 7x = 1 \]

### Step 7: Solve for \(x\)
Dividing both sides by \(7\):
\[ x = \frac{1}{7} \]

Thus, the value of \(x\) is \(\frac{1}{7}\).

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