$27x\sqrt{ x }=1$
Square both sides of the equation$729{x}^{2} \times x=1$
Calculate the product$729{x}^{3}=1$
Divide both sides of the equation by $729$${x}^{3}=\frac{ 1 }{ 729 }$
Take the root of both sides of the equation$x=\frac{ 1 }{ 9 }$
Check if the given value is the solution of the equation$27\sqrt{ \frac{ 1 }{ 9 } }=\frac{ 1 }{ \frac{ 1 }{ 9 } }$
Simplify the expression$9=9$
The equality is true, therefore $x=\frac{ 1 }{ 9 }$ is a solution of the equation$\begin{align*}&x=\frac{ 1 }{ 9 } \\&\begin{array} { l }x=0.\overset{ \cdot }{ 1 } ,& x={3}^{-2}\end{array}\end{align*}$