1. **State the problem:** Simplify the expression $$\left(\sqrt[5]{x^{4} y^{7}}\right)^{10}$$ and express it in the form $$x^{a} y^{b}$$ where $a$ and $b$ are exponents to find. 2. **Recall the rules:** - The fifth root of a quantity is the same as raising it to the power $\frac{1}{5}$. - When raising a power to another power, multiply the exponents: $\left(x^{m}\right)^{n} = x^{m \cdot n}$. 3. **Rewrite the expression using fractional exponents:** $$\left(x^{4} y^{7}\right)^{\frac{10}{5}} = \left(x^{4} y^{7}\right)^{2}$$ 4. **Apply the power to each factor inside the parentheses:** $$x^{4 \cdot 2} y^{7 \cdot 2} = x^{8} y^{14}$$ 5. **Final simplified form:** $$x^{8} y^{14}$$ So, the exponents are $8$ for $x$ and $14$ for $y$.