Simplifying Radicals

$\sqrt[9]{x^3}=\left (x^3\right)^{\frac{1}{9}}=x^{\frac{3}{9}}=x^{\frac{1}{3}}=\boxed{\sqrt[3] {x}}$

$\color{#3D99F6} \sqrt[9] {x^3}\color{#BA33D6}=\color{#D61F06}\sqrt[3] {\sqrt[3] {x^3}}\color{cyan}=\color{#20A900} \sqrt[3] x$ .

$\Large{\sqrt[9]{x^3} = x^{\frac{3}{9}} = x^{\frac{1}{3}} = \sqrt[3]{x}}$

Using the law of indices,

$\large \sqrt[9]{x^3} = (x^3)^{\frac19} = x^{3 \cdot \frac19} = x^{3/9} = x^{1/3} = \boxed{\sqrt[3]{x}} .$