1. The problem asks for the 5th root of $7^{10}$. 2. The formula for the $n$th root of a number $a^m$ is $$\sqrt[n]{a^m} = a^{\frac{m}{n}}$$. 3. Applying this formula, we get $$\sqrt[5]{7^{10}} = 7^{\frac{10}{5}}$$. 4. Simplify the exponent: $$7^{\frac{10}{5}} = 7^2$$. 5. Calculate $7^2$: $$7^2 = 49$$. 6. Therefore, the 5th root of $7^{10}$ is 49. Final answer: 49