Solve Nested Square Roots: √√2 × √√4 Product Evaluation

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

• Step 1: Simplify each term individually.
• Step 2: Multiply the simplified terms together.
• Step 3: Compare with choices if necessary.

Let's begin:

Step 1: Simplify each term:

The expression is $\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{4}}$.

- Simplifying $\sqrt{\sqrt{2}}$: A root of a root involves multiplying the indices. We have $\sqrt[2]{\sqrt[2]{2}}$, which becomes $\sqrt[4]{2}$.

- Simplifying $\sqrt{\sqrt{4}}$: Note that $\sqrt{4} = 2$, so $\sqrt{2} = \sqrt{2}$.

Conclusively, $\sqrt{\sqrt{4}} = \sqrt{2}$.

Step 2: Multiply the simplified terms:

Now, multiply $\sqrt[4]{2} \times \sqrt{2}$:

$\sqrt[4]{2} \cdot \sqrt{2} = \sqrt[4]{2 \times 2^{1/2}} = \sqrt[4]{2^{3/2}} = \sqrt[4]{8}$.

Therefore, our simplified expression is $\sqrt[4]{8}$.

Step 3: Compare with answer choices:

The correct choice is $\sqrt[4]{8}$, matching choice 3.

Therefore, the solution to the problem is $\sqrt[4]{8}$.