The rule of exponents:
[tex]\begin{gathered} ^m\sqrt[]{x^n}=x^{\frac{n}{m}} \\ x^n=\frac{1}{x^{-n}} \end{gathered}[/tex]
The given expression is,
[tex]^3\sqrt[]{\frac{x^6}{x^{-15}}}[/tex]
Rewrite the above expression as an exponential expression,
[tex]\begin{gathered} (\frac{x^6}{x^{-15}})^{\frac{1}{3}} \\ =(x^6x^{15})^{\frac{1}{3}} \\ =(x^{6+15})^{\frac{1}{3}} \\ =(x^{21})^{\frac{1}{3}} \\ =x^{\frac{21}{3}} \\ =x^7 \end{gathered}[/tex]
Therefore, the simplest exponential form of the given expression is,
[tex]x^7[/tex]
