Solve Nested Square Roots: √√16 × √√8 Multiplication Problem

To solve the problem $\sqrt{\sqrt{16}}\cdot\sqrt{\sqrt{8}}$, we will follow these steps:

• Step 1: Simplify each nested square root expression.
• Step 2: Multiply the simplified expressions of the roots.

Step 1: Evaluate $\sqrt{\sqrt{16}}$.
Since 16 can be expressed as $2^4$, we have: $\sqrt{\sqrt{16}} = \left(16^{1/2}\right)^{1/2} = 16^{1/4} = \left(2^4\right)^{1/4} = 2^{4/4} = 2^{1} = 2$

Evaluate $\sqrt{\sqrt{8}}$.
Since 8 can be expressed as $2^3$, we have: $\sqrt{\sqrt{8}} = \left(8^{1/2}\right)^{1/2} = 8^{1/4} = \left(2^3\right)^{1/4} = 2^{3/4}$

Step 2: Multiply these simplified expressions together: $2 \cdot 2^{3/4} = 2^{1} \cdot 2^{3/4} = 2^{1 + 3/4} = 2^{7/4}$

Finally, converting back to radical form: $2^{7/4} = \left(2^7\right)^{1/4} = \sqrt[4]{2^7} = \sqrt[4]{128}$

Thus, the solution to the problem is $\sqrt[4]{128}$, which corresponds to answer choice 3.