Solve for x :
`(root3((3)/(5)))^(2x+2)=(25)/(9)`

Answer

To solve the equation \( \left(\sqrt[3]{\frac{3}{5}}\right)^{2x+2} = \frac{25}{9} \), we will follow these steps:

### Step 1: Rewrite the Cube Root
We can rewrite the cube root as an exponent:
\[ \sqrt[3]{\frac{3}{5}} = \left(\frac{3}{5}\right)^{\frac{1}{3}} \]
Thus, the equation becomes:
\[ \left(\frac{3}{5}\right)^{\frac{1}{3}(2x+2)} = \frac{25}{9} \]

### Step 2: Simplify the Left Side
Using the property of exponents \( a^{m \cdot n} = a^{mn} \):
\[ \left(\frac{3}{5}\right)^{\frac{2x+2}{3}} = \frac{25}{9} \]

### Step 3: Rewrite the Right Side
We can express \( \frac{25}{9} \) in terms of base \( \frac{5}{3} \):
\[ \frac{25}{9} = \left(\frac{5}{3}\right)^2 \]

### Step 4: Equate the Bases
Now, we have:
\[ \left(\frac{3}{5}\right)^{\frac{2x+2}{3}} = \left(\frac{5}{3}\right)^2 \]
We can rewrite \( \left(\frac{5}{3}\right)^2 \) as \( \left(\frac{3}{5}\right)^{-2} \):
\[ \left(\frac{3}{5}\right)^{\frac{2x+2}{3}} = \left(\frac{3}{5}\right)^{-2} \]

### Step 5: Set the Exponents Equal
Since the bases are equal, we can set the exponents equal to each other:
\[ \frac{2x+2}{3} = -2 \]

### Step 6: Solve for \( x \)
Multiply both sides by 3 to eliminate the fraction:
\[ 2x + 2 = -6 \]
Now, subtract 2 from both sides:
\[ 2x = -8 \]
Finally, divide by 2:
\[ x = -4 \]

### Final Answer
Thus, the solution for \( x \) is:
\[ \boxed{-4} \]