Solve for \( x \) : \( \log _{3}\left(x^{9}\right)=-2 \) \( x=\square \) You may enter the exact value or round to 4 decimal places. Question Help: Dideo Submit Ouestion

To solve the equation \( \log _{3}\left(x^{9}\right)=-2 \), we can rewrite it in exponential form. This gives us: \[ x^{9} = 3^{-2} \] Since \( 3^{-2} = \frac{1}{3^{2}} = \frac{1}{9} \), we can rewrite the equation as: \[ x^{9} = \frac{1}{9} \] Now, to solve for \( x \), we take the ninth root of both sides: \[ x = \left(\frac{1}{9}\right)^{\frac{1}{9}} \] This simplifies to: \[ x = \frac{1}{9^{\frac{1}{9}}} \] As \( 9 = 3^{2} \), we can express it as: \[ 9^{\frac{1}{9}} = (3^{2})^{\frac{1}{9}} = 3^{\frac{2}{9}} \] Thus, \[ x = \frac{1}{3^{\frac{2}{9}}} = 3^{-\frac{2}{9}} \] Calculating this gives us the approximate value of \( x \): \[ x \approx 0.4395 \] So the answer is: \( x \approx 0.4395 \)