Let's solve the problem step-by-step logically and systematically to find out the square root of [tex]\(2^{1/4}\)[/tex] from the given options:
1. We start by calculating the expression [tex]\(2^{1/4}\)[/tex], which is the fourth root of 2. We represent this mathematically as:
[tex]\[
2^{1/4} = \sqrt[4]{2}
\][/tex]
2. Next, we need to find the square root of this result. If we denote the fourth root of 2 as [tex]\(x\)[/tex], then:
[tex]\[
x = 2^{1/4}
\][/tex]
3. We now take the square root of [tex]\(x\)[/tex]:
[tex]\[
\sqrt{2^{1/4}} = (2^{1/4})^{1/2}
\][/tex]
4. Using the properties of exponents, specifically [tex]\( (a^m)^n = a^{m \cdot n} \)[/tex], we simplify the exponent:
[tex]\[
(2^{1/4})^{1/2} = 2^{(1/4) \cdot (1/2)} = 2^{1/8}
\][/tex]
5. We need to evaluate which of the given options corresponds to [tex]\(2^{1/8}\)[/tex]. The options are:
[tex]\[
\text{(a)} \ \frac{1}{8}, \quad \text{(b)} \ \frac{1}{4}, \quad \text{(c)} \ \frac{1}{2}, \quad \text{(d)} \ \frac{1}{4}, \quad \text{(e)} \ \frac{11}{2}
\][/tex]
6. Since none of these options directly represent [tex]\(2^{1/8}\)[/tex] as a straightforward fraction, and considering floating-point comparison inaccuracies, we go through the options to see if [tex]\(2^{1/8}\)[/tex] closely matches any of them.
Checking each option logically:
- [tex]\( \frac{1}{8} = 0.125 \)[/tex]
- [tex]\( \frac{1}{4} = 0.25 \)[/tex]
- [tex]\( \frac{1}{2} = 0.5 \)[/tex]
- [tex]\( \frac{11}{2} = 5.5 \)[/tex]
Comparing these values with [tex]\(2^{1/8}\)[/tex], which is approximately [tex]\(1.0905\)[/tex], none of the given options match [tex]\(2^{1/8}\)[/tex].
Therefore, the correct conclusion based on the available options is:
None of the given options correspond to the square root of [tex]\(2^{1/4}\)[/tex].
Thus, the answer is:
[tex]\[ \boxed{\text{None}} \][/tex]
