Solve the Nested Root Expression: Fifth Root of Cube Root of 5

To solve the problem of finding $\sqrt[5]{\sqrt[3]{5}}$, we'll use the formula for a root of a root, which combines the exponents:

• Step 1: Express each root as an exponent.
  We start with the innermost root: $\sqrt[3]{5} = 5^{1/3}$.
• Step 2: Apply the outer root.
  The square root to the fifth power is expressed as: $\sqrt[5]{5^{1/3}} = (5^{1/3})^{1/5}$.
• Step 3: Combine the exponents.
  Using the exponent rule $(a^m)^n = a^{m \times n}$, we get:
  $(5^{1/3})^{1/5} = 5^{(1/3) \times (1/5)} = 5^{1/15}$.
• Step 4: Convert the exponent back to root form.
  This can be written as $\sqrt[15]{5}$.

Therefore, the simplified expression of $\sqrt[5]{\sqrt[3]{5}}$ is $\sqrt[15]{5}$.