Solve Nested Roots: Finding Fifth Root of Square Root of 1024

To solve this problem, we'll follow these steps:

• Step 1: Simplify the inner square root.
• Step 2: Apply the formula for nested roots.
• Step 3: Compute the fifth root.

Now, let's work through each step:

Step 1: Calculate the inner square root.
We have $\sqrt{1024}$. We know that $1024 = 2^{10}$, so $\sqrt{1024} = \sqrt{2^{10}}$.

Applying the property of roots, $\sqrt{2^{10}} = 2^{\frac{10}{2}} = 2^5 = 32$.

Step 2: Now, apply the fifth root to the result from step 1.
We need to find $\sqrt[5]{32}$.

Step 3: Simplify using the properties of exponents.
From $\sqrt[5]{32} = \sqrt[5]{2^5}$, we have $2^{\frac{5}{5}} = 2^1 = 2$.

Therefore, the solution to the problem is $2$.