Simplify the \( f \) (1) \( \frac{\left(27 x^{9}\right)}{\sqrt{9^{-1}}} \)

1. Rewrite the expression: \[ \frac{27 x^{9}}{\sqrt{9^{-1}}} \] 2. Evaluate the square root in the denominator. Notice that: \[ \sqrt{9^{-1}} = \left(9^{-1}\right)^{\frac{1}{2}} = 9^{-\frac{1}{2}} \] 3. Since \(9^{\frac{1}{2}} = 3\), we have: \[ 9^{-\frac{1}{2}} = \frac{1}{9^{\frac{1}{2}}} = \frac{1}{3} \] 4. Substitute back into the original expression: \[ \frac{27 x^{9}}{\frac{1}{3}} = 27 x^{9} \times 3 \] 5. Multiply: \[ 27 x^{9} \times 3 = 81 x^{9} \] Thus, the simplified expression is: \[ 81 x^{9} \]