Solve the Nested Radical: Seventh Root of Square Root of 2

Let's solve the given problem by following these steps:

• Step 1: Recognize the expression $\sqrt[7]{\sqrt{2}}$. It involves two roots.
• Step 2: Rewrite each part using rational exponents. We have $\sqrt{2} = 2^{1/2}$.
• Step 3: Substitute back, giving $\sqrt[7]{2^{1/2}}$ or $(2^{1/2})^{1/7}$.
• Step 4: Use the properties of exponents: $(a^m)^n = a^{m \cdot n}$.
• Step 5: Calculate the exponent: $(1/2) \cdot (1/7) = 1/14$.
• Step 6: This gives us $2^{1/14}$, which is equal to $\sqrt[14]{2}$.

Thus, the simplified expression is $\sqrt[14]{2}$.

Therefore, the solution to the problem is $\sqrt[14]{2}$.