Let's start by understanding the problem statement. We are given a statement:
[tex]\[
\sqrt[3]{2} = 2^{\frac{1}{4}}
\][/tex]
and we need to find the correct simplification from the given options.
Let's examine each option step-by-step:
### Option A
[tex]\[
\left(2^{\frac{1}{6}}\right)^4 = 2^{\frac{1}{4}} \cdot 2^{\frac{1}{6}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}}
\][/tex]
From this point, the process becomes somewhat unclear. The expression doesn't follow the proper rules for combining exponents and has an incorrect intermediate step involving multiplication that doesn't follow logically:
[tex]\[
4 \cdot 2^{\frac{1}{4}} = 4 \cdot \frac{1}{4} \cdot 2
\][/tex]
Thus, this option is incorrect.
### Option B
[tex]\[
\left(2^{\frac{1}{6}}\right)^4 = 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}}
\][/tex]
Appropriate multiplication rules apply here:
[tex]\[
2 \cdot\left(\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}\right) = 2 \cdot \frac{4}{4}
\][/tex]
Once again, an incorrect step appears when combining terms improperly:
[tex]\[
= 2
\][/tex]
This interpretation is also flawed.
### Option C
[tex]\[
\left(2^{\frac{1}{6}}\right)^4 = 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}}
\][/tex]
Here, simplifying the exponents, we add the powers:
[tex]\[
2^{\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}} = 2^{\frac{4}{4}} = 2^1 = 2
\][/tex]
This option follows correctly and reaches the conclusion directly and simply.
### Option D
[tex]\[
\left(2^{\frac{1}{4}}\right)^4 = 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}}
\][/tex]
Simplifying within the exponents yields:
[tex]\[
2^{\frac{1}{4} \cdot \frac{1}{4} \cdot \frac{1}{4} \cdot \frac{1}{4}}
\][/tex]
This combines exponents incorrectly, leading to misleading results:
[tex]\[
2^{\frac{4}{4}} = 2^1 = 2
\][/tex]
But, this contains an improper exponent manipulation.
Hence, the correct simplification of the given expression is accurately reflected in:
### Answer: C
