Solve the Nested Radical: Simplifying the Sixth Root of Square Root of 2

Solve the following exercise:

$\sqrt[6]{\sqrt{2}}=$

In order to solve this problem, we must simplify the following expression $\sqrt[6]{\sqrt{2}}$ using the rule for roots of roots. This rule states that a root of a root can be written as a single root by multiplying the indices of the radicals.

• Step 1: Identify the given expression $\sqrt[6]{\sqrt{2}}$.

• Step 2: Recognize that the inner root, $\sqrt{2}$, can be expressed as $\sqrt[2]{2}$.

• Step 3: Visualize $\sqrt[6]{\sqrt{2}}$ as $\sqrt[6]{\sqrt[2]{2}}$.

• Step 4: Apply the rule $\sqrt[n]{\sqrt[m]{a}} = \sqrt[n \times m]{a}$.

• Step 5: Multiply the indices: $6 \times 2 = 12$.

• Step 6: Replace the compound root with the single root: $\sqrt[12]{2}$.

Thus, the expression $\sqrt[6]{\sqrt{2}}$ simplifies to $\sqrt[12]{2}$.

Therefore, the solution to the problem is $\sqrt[12]{2}$.