To simplify \(\sqrt{52x^4}\), we start by breaking down the expression under the square root.
First, we can factor \(52\) into its prime factors: \[ 52 = 4 \cdot 13 = 2^2 \cdot 13 \] Next, we rewrite \(x^4\) in terms of perfect squares: \[ x^4 = (x^2)^2 \] Now, we can rewrite \(\sqrt{52x^4}\) as: \[ \sqrt{52x^4} = \sqrt{52} \cdot \sqrt{x^4} \]
Substituting the factored forms, we have: \[ \sqrt{52} = \sqrt{4 \cdot 13} = \sqrt{4} \cdot \sqrt{13} = 2\sqrt{13} \] and \[ \sqrt{x^4} = x^2 \] Thus, we combine these results: \[ \sqrt{52x^4} = 2\sqrt{13} \cdot x^2 \]
Therefore, the final simplified expression is: \[ \boxed{2x^2\sqrt{13}} \]
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