To simplify the given expression, we need to identify the simplification of the powers of 3 that are provided in the choices. The given options are:
1. [tex]\(3^{\frac{1}{20}}\)[/tex]
2. [tex]\(3^{\frac{9}{20}}\)[/tex]
3. [tex]\(3^{\frac{5}{4}}\)[/tex]
4. [tex]\(3^{\frac{1}{5}}\)[/tex]
Let's look at each option and consider what it represents:
1. [tex]\(3^{\frac{1}{20}}\)[/tex] - This represents 3 raised to the power of [tex]\(\frac{1}{20}\)[/tex]. This is a very small exponent and does not simplify to a common or familiar value.
2. [tex]\(3^{\frac{9}{20}}\)[/tex] - This represents 3 raised to the power of [tex]\(\frac{9}{20}\)[/tex].
3. [tex]\(3^{\frac{5}{4}}\)[/tex] - This can be rewritten as [tex]\((3^{\frac{1}{4}})^5\)[/tex]. While this could be simplified, it does not match any easily recognizable simplification pattern.
4. [tex]\(3^{\frac{1}{5}}\)[/tex] - This represents 3 raised to the power of [tex]\(\frac{1}{5}\)[/tex].
While each of these options are valid forms, they do not simplify in a way that reduces them to a common simpler numeric form.
Given the choices, and the complexity of each fractional exponent, each of these forms represents a unique value that does not reduce further by common mathematical simplification rules.
Therefore, the given answer choices are already in their simplest forms.
