To solve the given problem, we need to follow the steps of simplification for the expression [tex]\(\left(2^{\frac{1}{4}}\right)^4\)[/tex].
Let's evaluate this step by step:
1. [tex]\(\left(2^{\frac{1}{4}}\right)^4\)[/tex].
2. When raising a power to another power, the exponents get multiplied:
[tex]\[
\left(a^{m}\right)^{n} = a^{m \cdot n}
\][/tex]
Applying this property, we get:
[tex]\[
\left(2^{\frac{1}{4}}\right)^{4} = 2^{\frac{1}{4} \cdot 4}
\][/tex]
3. Simplifying the exponent:
[tex]\[
2^{\frac{1}{4} \cdot 4} = 2^{1}
\][/tex]
4. Any number to the power of 1 is the number itself:
[tex]\[
2^{1} = 2
\][/tex]
Therefore, the correct simplification according to the rules of exponents is [tex]\(2\)[/tex].
Checking the provided options:
A. [tex]\(\left(2^{\frac{1}{4}}\right)^4=2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}}=2^{\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}}=2^{\frac{4}{4}}=2^1=2\)[/tex]
This is accurately done, so this is correct.
B. Using addition inside the base of 2, that leads to confusion and is incorrect.
C. Product of fractions rather than summation is applied incorrectly and is incorrect.
D. Incorrectly uses direct multiplication without correct exponent rules, is also incorrect.
The accurate simplification is given by option A. Therefore, the correct answer is:
[tex]\[
\boxed{A}
\][/tex]
