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Sagot :
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]
[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]
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