Question Solve the following equation for \( x \). \[ (x+2)^{\frac{4}{3}}=\frac{1}{81} \] Assume \( x+2 \) is positive. Give your answer as a simplified integer or improper fraction, if necessary. Provide your answer below:

To solve the equation \( (x+2)^{\frac{4}{3}}=\frac{1}{81} \), we start by raising both sides to the power of \(\frac{3}{4}\) to eliminate the exponent on the left side. This gives us: \[ x + 2 = \left(\frac{1}{81}\right)^{\frac{3}{4}}. \] Next, we need to simplify \(\left(\frac{1}{81}\right)^{\frac{3}{4}}\). Since \(81\) can be expressed as \(3^4\), we have: \[ \frac{1}{81} = 3^{-4}, \] so: \[ \left(\frac{1}{81}\right)^{\frac{3}{4}} = (3^{-4})^{\frac{3}{4}} = 3^{-3} = \frac{1}{27}. \] Now we can substitute this back into our equation: \[ x + 2 = \frac{1}{27}. \] To isolate \(x\), we subtract \(2\) from both sides: \[ x = \frac{1}{27} - 2. \] To combine these terms, we can express \(2\) as \(\frac{54}{27}\): \[ x = \frac{1}{27} - \frac{54}{27} = \frac{1 - 54}{27} = \frac{-53}{27}. \] Thus, the solution is: \[ \boxed{\frac{-53}{27}}. \]