Root Power 171De0

1. **State the problem:** Simplify the expression $$\left(\sqrt[5]{2p}\right)^8$$. 2. **Recall the formula:** The nth root of a number $a$ can be written as a fractional exponent: $$\sqrt[n]{a} = a^{\frac{1}{n}}$$. 3. **Rewrite the expression:** Using this rule, $$\left(\sqrt[5]{2p}\right)^8 = \left((2p)^{\frac{1}{5}}\right)^8$$. 4. **Apply the power of a power rule:** When raising a power to another power, multiply the exponents: $$\left((2p)^{\frac{1}{5}}\right)^8 = (2p)^{\frac{1}{5} \times 8} = (2p)^{\frac{8}{5}}$$. 5. **Final answer:** The simplified form is $$\boxed{(2p)^{\frac{8}{5}}}$$. This means the expression is the quantity $2p$ raised to the power $\frac{8}{5}$.