The problem asks to express $\sqrt[3]{\sqrt[5]{5}}$ as a single radical using the smallest possible root.

Let's break it down:

1. Start with $\sqrt[5]{5}$, which means $5^{1/5}$.
2. Now, take the cube root of this expression: [ \sqrt[3]{\sqrt[5]{5}} = \sqrt[3]{5^{1/5}} = \left( 5^{1/5} \right)^{1/3} ]
3. Using the property of exponents $(a^m)^n = a^{m \cdot n}$, we get: $5^{(1/5) \cdot (1/3)} = 5^{1/15}$

Thus, $\sqrt[3]{\sqrt[5]{5}} = 5^{1/15}$, which can be written as $\sqrt[15]{5}$.

Final answer:

[ \sqrt[3]{\sqrt[5]{5}} = \sqrt[15]{5} ]

Would you like more details on this process, or do you have any questions?

Related questions:

1. How do you simplify expressions with nested radicals in general?
2. What are the key properties of exponents needed to simplify radicals?
3. How do fractional exponents relate to radicals?
4. Can you convert any radical into an expression with exponents?
5. What is the difference between cube roots and fifth roots in practical applications?

Tip:

Always simplify nested radicals by converting them into fractional exponents; this makes it easier to combine them into a single radical expression.