Solve Nested Square Roots: √√4 × √√2 Multiplication Problem

To solve the expression $\sqrt{\sqrt{4}}\cdot\sqrt{\sqrt{2}}$, we will use properties of exponents and roots.

First, let's simplify each part:

• $\sqrt{\sqrt{4}}$:

We know $\sqrt{4} = 2$. Therefore, $\sqrt{\sqrt{4}}$ can be rewritten as $\sqrt{2}$, because $\sqrt{4} = 2$ and further taking square root gives $2^{1/2}$.

• $\sqrt{\sqrt{2}}$:

This expression is equivalent to $(2^{1/2})^{1/2}$. Using the property $(a^{m})^{n} = a^{m \cdot n}$, we have:

$(2^{1/2})^{1/2} = 2^{1/4}$.

Now, the original expression simplifies to:

$\sqrt{2} \cdot 2^{1/4}$

This product is expressed as:

$2^{1/2} \cdot 2^{1/4}$. When multiplying like bases, add the exponents:

$2^{1/2 + 1/4} = 2^{2/4 + 1/4} = 2^{3/4}$

Thus, the final expression is:

$2^{1/4}\sqrt{2}$.

Comparing this to the choices provided, the correct answer is:

$2^{1/4}\sqrt{2}$ (Choice 3).

Therefore, the solution to the problem is $\boxed{2^{\frac{1}{4}}\sqrt{2}}$.