Solve: Multiplication of Cube Roots with Nested Square Roots (√[3]{√3}·√[3]{√4})

To solve $\sqrt[3]{\sqrt{3}} \cdot \sqrt[3]{\sqrt{4}}$, we will convert these expressions into powers:

Step 1: Express each root as a power:
$\sqrt[3]{\sqrt{3}} = (\sqrt{3})^{1/3} = 3^{1/6}$
$\sqrt[3]{\sqrt{4}} = (\sqrt{4})^{1/3} = 4^{1/6}$

Step 2: Multiply the expressions using the property of exponents:
$3^{1/6} \cdot 4^{1/6} = (3 \cdot 4)^{1/6} = 12^{1/6}$

Therefore, the simplified expression is $\sqrt[6]{12}$.