\( \sqrt {3}.\sqrt {48}-\frac{4^{x+1}}{2^{2x}} \)

To simplify the expression \( \sqrt{3} \cdot \sqrt{48} - \frac{4^{x+1}}{2^{2x}} \), let’s break it down step by step. First, simplify \( \sqrt{48} \): \[ \sqrt{48} = \sqrt{16 \cdot 3} = \sqrt{16} \cdot \sqrt{3} = 4\sqrt{3}. \] Now, substitute back into the expression: \[ \sqrt{3} \cdot 4\sqrt{3} = 4(\sqrt{3})^2 = 4 \cdot 3 = 12. \] Next, simplify \( \frac{4^{x+1}}{2^{2x}} \): We know that \( 4 = 2^2 \), so: \[ 4^{x+1} = (2^2)^{x+1} = 2^{2(x+1)} = 2^{2x + 2}. \] Now, substitute this into the expression: \[ \frac{2^{2x + 2}}{2^{2x}} = 2^{(2x + 2) - 2x} = 2^2 = 4. \] Putting it all together, we have: \[ 12 - 4 = 8. \] Thus, the simplified form of the expression is \( 8 \).