Find x, if :
`(root(3)((2)/(3)))^(x-1)=(27)/(8)`

Answer

To solve the equation \((\sqrt[3]{\frac{2}{3}})^{x-1} = \frac{27}{8}\), we can follow these steps:

### Step 1: Rewrite the cube root
The cube root can be expressed as an exponent:
\[ \sqrt[3]{\frac{2}{3}} = \left(\frac{2}{3}\right)^{\frac{1}{3}} \]
Thus, we can rewrite the left side of the equation:
\[ \left(\left(\frac{2}{3}\right)^{\frac{1}{3}}\right)^{x-1} = \frac{27}{8} \]

### Step 2: Apply the power of a power property
Using the property \((a^m)^n = a^{m \cdot n}\), we can simplify the left side:
\[ \left(\frac{2}{3}\right)^{\frac{x-1}{3}} = \frac{27}{8} \]

### Step 3: Rewrite the right side
Next, we rewrite \(\frac{27}{8}\) in terms of powers:
\[ 27 = 3^3 \quad \text{and} \quad 8 = 2^3 \]
Thus, we can express \(\frac{27}{8}\) as:
\[ \frac{27}{8} = \frac{3^3}{2^3} = \left(\frac{3}{2}\right)^3 \]

### Step 4: Set the bases equal
Now we have:
\[ \left(\frac{2}{3}\right)^{\frac{x-1}{3}} = \left(\frac{3}{2}\right)^3 \]
We can rewrite \(\frac{3}{2}\) as \(\left(\frac{2}{3}\right)^{-1}\):
\[ \left(\frac{2}{3}\right)^{\frac{x-1}{3}} = \left(\frac{2}{3}\right)^{-3} \]

### Step 5: Equate the exponents
Since the bases are equal, we can set the exponents equal to each other:
\[ \frac{x-1}{3} = -3 \]

### Step 6: Solve for \(x\)
To solve for \(x\), multiply both sides by 3:
\[ x - 1 = -9 \]
Now, add 1 to both sides:
\[ x = -9 + 1 = -8 \]

### Final Answer
Thus, the value of \(x\) is:
\[ \boxed{-8} \]