Select the correct answer.

Select the simplification that accurately explains the following statement.
[tex]\sqrt[3]{2}=2^{\frac{1}{4}}[/tex]

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}}=4 \cdot 2^{\frac{1}{4}}=4 \cdot \frac{1}{4} \cdot 2=2[/tex]

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}}=2 \cdot\left(\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}\right)=2 \cdot \frac{4}{4}=2[/tex]

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}}=2^{\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}}=2^{\frac{4}{4}}=2^1=2[/tex]

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}}=2^{\frac{1}{4} \cdot \frac{1}{4} \cdot \frac{1}{4} \cdot \frac{1}{4}}=2^{\frac{4}{4}}=2^1=2[/tex]



Answer :

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