To simplify the given radical [tex]\(\sqrt{\frac{19}{x^4}}\)[/tex], we start by separating the expressions under the square root.
[tex]\[
\sqrt{\frac{19}{x^4}} = \sqrt{19} \cdot \sqrt{\frac{1}{x^4}}
\][/tex]
Next, we can simplify [tex]\(\sqrt{\frac{1}{x^4}}\)[/tex]. Recall that [tex]\(\sqrt{\frac{1}{x^4}} = \sqrt{x^{-4}}\)[/tex], which simplifies further:
[tex]\[
\sqrt{x^{-4}} = x^{-2}
\][/tex]
This is because taking the square root of [tex]\(x^{-4}\)[/tex] changes the exponent from [tex]\(-4\)[/tex] to [tex]\(-2\)[/tex]. Therefore, we now have:
[tex]\[
\sqrt{\frac{19}{x^4}} = \sqrt{19} \cdot x^{-2}
\][/tex]
Since [tex]\(x^{-2} = \frac{1}{x^2}\)[/tex], we can rewrite the expression as:
[tex]\[
\sqrt{19} \cdot \frac{1}{x^2} = \frac{\sqrt{19}}{x^2}
\][/tex]
Hence, the simplified form of the radical [tex]\(\sqrt{\frac{19}{x^4}}\)[/tex] is:
[tex]\[
\frac{\sqrt{19}}{x^2}
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{\frac{\sqrt{19}}{x^2}}
\][/tex]
