To simplify \(\sqrt{52x^4}\), we start by breaking it down into its factors.
First, we factor \(52\): \[ 52 = 4 \times 13 \] Next, we can recognize that \(4\) is a perfect square.
Now we can write: \[ \sqrt{52x^4} = \sqrt{4 \times 13 \times x^4} \] Using the property of square roots that states \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\), we can separate the factors under the square root: \[ \sqrt{52x^4} = \sqrt{4} \times \sqrt{13} \times \sqrt{x^4} \]
Next, we calculate each square root: \[ \sqrt{4} = 2 \] \[ \sqrt{x^4} = x^2 \quad (\text{since } \sqrt{x^n} = x^{n/2}) \]
Putting it all together, we have: \[ \sqrt{52x^4} = 2 \times \sqrt{13} \times x^2 \] Thus, we can combine this into: \[ \sqrt{52x^4} = 2x^2\sqrt{13} \]
Therefore, the final simplified expression is: \[ \boxed{2x^2\sqrt{13}} \]
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