To solve the given problem, we need to find the correct simplification for the expression [tex]\(\sqrt[4]{2} = 2^{\frac{1}{4}}\)[/tex].
Let's examine each option one by one to determine which one correctly simplifies the given expression.
Option A:
[tex]\[
\left(2^{\frac{1}{4}}\right)^4=2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}}=4 \cdot 2^{\frac{1}{4}}=4 \cdot \frac{1}{4} \cdot 2=2
\][/tex]
This option incorrectly states that multiplying four terms of [tex]\(2^{\frac{1}{4}}\)[/tex] results in [tex]\(4 \cdot 2^{\frac{1}{4}}\)[/tex], which is not correct. The exponentiation rule has not been properly applied here.
Option B:
[tex]\[
\left(2^{\frac{1}{4}}\right)^4=2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}}=2 \cdot\left(\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}\right)=2 \cdot \frac{4}{4}=2
\][/tex]
This option incorrectly treats the addition within the exponent as a multiplication factor outside, which is not a correct interpretation of exponent addition.
Option C:
[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]
This option incorrectly multiplies the exponents, while the correct rule should be to add the exponents when bases are multiplied.
Option D:
[tex]\[
\left(2^{\frac{1}{4}}\right)^4=2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}}=2^{\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}}=2^{\frac{4}{4}}=2^1=2
\][/tex]
This option correctly applies the exponentiation rule. When multiplying terms with the same base, we add the exponents:
[tex]\[
\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} = 1,
\][/tex]
so,
[tex]\[
2^{\frac{4}{4}} = 2^{1} = 2.
\][/tex]
Therefore, the correct answer is:
D. [tex]\(\left(2^{\frac{1}{4}}\right)^4=2^{\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}}=2^{\frac{4}{4}}=2^1=2.\)[/tex]
