\[\begin{array}{l}\frac{{{{\left( {{7^{\frac{1}{3}}} \cdot {7^{ - \frac{2}{3}}}} \right)}^3}}}{{{7^{ - 3}}}} = \frac{{{{\left( {{7^{\frac{1}{3}}}} \right)}^3} \cdot {{\left( {{7^{ - \frac{2}{3}}}} \right)}^3}}}{{{7^{ - 3}}}} = \frac{{7 \cdot {7^{ - 2}}}}{{{7^{ - 3}}}} = \\\frac{{{7^{ - 1}}}}{{{7^{ - 3}}}} = {7^{ - 1 - \left( { - 3} \right)}} = {7^{ - 1 + 3}} = {7^2} = 49\end{array}\]

\[49\]