Step 1 :The given expression is \(\sqrt{48 x^{12}}\).
Step 2 :To simplify this, we need to break down the number 48 into its prime factors and simplify the power of x.
Step 3 :The prime factorization of 48 is \(2^4 * 3\).
Step 4 :The power of x can be simplified by dividing the exponent by 2, since we are taking the square root.
Step 5 :The simplified expression is \(4\sqrt{3}x^6\). This is because the square root of \(x^{12}\) is \(x^6\) and the square root of 48 is \(4\sqrt{3}\).
Step 6 :Final Answer: \(\sqrt{48 x^{12}} = \boxed{4\sqrt{3}x^6}\)