Simplify.

Remove all perfect squares from inside the square root.
\[\sqrt{52x^4}=\]

User Icon for Bot GPT 4-o mini Bot GPT 4-o mini answered
2 hours ago

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:
\[
\boxed{2x^2\sqrt{13}}
\]