To determine the simplified form of [tex]\(\sqrt[5]{x}\)[/tex], we need to understand how to express it using exponents.
1. The radical form [tex]\(\sqrt[5]{x}\)[/tex] is a way to express the fifth root of [tex]\(x\)[/tex].
2. The fifth root of [tex]\(x\)[/tex] can be rewritten as a fractional exponent. In general, the [tex]\(n\)[/tex]-th root of a number [tex]\(a\)[/tex] can be rewritten as [tex]\(a^{\frac{1}{n}}\)[/tex].
3. Hence, for our specific case ([tex]\(\sqrt[5]{x}\)[/tex]), [tex]\(n\)[/tex] is 5. Therefore, the fifth root of [tex]\(x\)[/tex] can be rewritten as:
[tex]\[
x^{\frac{1}{5}}
\][/tex]
Now let's examine the given options to see which matches this simplified form:
1. [tex]\(x^{\frac{1}{5}}\)[/tex] - This matches the form we derived ([tex]\(x^{\frac{1}{5}}\)[/tex]).
2. [tex]\(x^{\frac{4}{5}}\)[/tex] - This is not the correct representation of [tex]\(\sqrt[5]{x}\)[/tex] because the exponent is [tex]\(\frac{4}{5}\)[/tex] instead of [tex]\(\frac{1}{5}\)[/tex].
3. [tex]\(x^{\frac{4}{20}}\)[/tex] - This can be simplified to [tex]\(x^{\frac{1}{5}}\)[/tex], but to stay consistent with simplicity and clarity, we prefer [tex]\(x^{\frac{1}{5}}\)[/tex] directly.
4. [tex]\(x\)[/tex] - This represents [tex]\(x\)[/tex] itself and not its fifth root.
Therefore, the simplified form of [tex]\(\sqrt[5]{x}\)[/tex] is:
[tex]\[
x^{\frac{1}{5}}
\][/tex]
The correct answer is option 1.
