1. **Problem statement:** Simplify the expression $$\sqrt[10]{a^7} : \sqrt[5]{a^3}$$. 2. **Recall the rule for radicals and exponents:** $$\sqrt[n]{a^m} = a^{\frac{m}{n}}$$ 3. **Rewrite each radical as an exponent:** $$\sqrt[10]{a^7} = a^{\frac{7}{10}}$$ $$\sqrt[5]{a^3} = a^{\frac{3}{5}}$$ 4. **Division of powers with the same base:** $$a^{\frac{7}{10}} : a^{\frac{3}{5}} = a^{\frac{7}{10} - \frac{3}{5}}$$ 5. **Find common denominator and subtract exponents:** $$\frac{3}{5} = \frac{6}{10}$$ $$a^{\frac{7}{10} - \frac{6}{10}} = a^{\frac{1}{10}}$$ 6. **Final simplified form:** $$a^{\frac{1}{10}} = \sqrt[10]{a}$$ **Answer:** $$\sqrt[10]{a}$$