Solve for \( x \) : \( \log _{3}\left(x^{9}\right)=-7 \) \( x=\square \) You may enter the exact value or round to 4 decimal places.

To solve for \( x \) in the equation \( \log_{3}(x^{9}) = -7 \), we can rewrite the logarithmic equation in its exponential form: \[ x^{9} = 3^{-7} \] Now, calculate \( 3^{-7} \): \[ 3^{-7} = \frac{1}{3^{7}} = \frac{1}{2187} \] Next, we need to solve for \( x \) by taking the 9th root of both sides: \[ x = \left(\frac{1}{2187}\right)^{\frac{1}{9}} \] Calculating the 9th root of \( \frac{1}{2187} \) gives us: \[ x = \frac{1}{(2187)^{\frac{1}{9}}} \] Now we find \( 2187^{\frac{1}{9}} \). The value of \( 2187 \) can be written as \( 3^7 \), thus: \[ (2187)^{\frac{1}{9}} = (3^7)^{\frac{1}{9}} = 3^{\frac{7}{9}} \] So, we have: \[ x = \frac{1}{3^{\frac{7}{9}}} = 3^{-\frac{7}{9}} \] This can also be approximated: Calculating \( 3^{-\frac{7}{9}} \) numerically, we find: \[ x \approx 0.2177 \quad \text{(rounded to four decimal places)} \] Thus, the solution is: \[ x \approx 0.2177 \]