To solve the problem of simplifying [tex]\(\left(2^{\frac{1}{4}}\right)^4\)[/tex], let's consider each option step-by-step:
Option A:
[tex]\[
\left(2^{\frac{1}{4}}\right)^4 = 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}}
\][/tex]
Using the property of exponents [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]:
[tex]\[
= 2^{\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4}}
= 2^{\frac{4}{4}}
= 2^1
= 2
\][/tex]
This looks correct.
Option B:
[tex]\[
\left(2^{\frac{1}{4}}\right)^4 = 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}}
\][/tex]
Using the same property of exponents, we should add the exponents, not multiply them:
[tex]\[
2^{\frac{1}{4} \cdot \frac{1}{4} \cdot \frac{1}{4} \cdot \frac{1}{4}} = 2^{\frac{1}{4 + 1/4 + 1/4 + 1/4}} = 2^{1} = 2
\][/tex]
But the step [tex]\(\frac{1}{4} \cdot \frac{1}{4} \cdot \frac{1}{4} \cdot \frac{1}{4}\)[/tex] is incorrect because it should be addition, not multiplication. This option is incorrect.
Option C:
[tex]\[
\left(2^{\frac{1}{4}}\right)^4 = 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}}
\][/tex]
Incorrectly, this option assumes:
[tex]\[
= 4 \cdot 2^{\frac{1}{4}} \quad (\text{wrong})
= 4 \cdot \frac{1}{4} \cdot 2 = 2 \quad (\text{wrong principles applied})
\][/tex]
This step does not follow the property of exponents. It's incorrect.
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]
Again, using the property of exponents [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]:
[tex]\[
= 2^{\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4}}
= 2^{\frac{4}{4}}
= 2^1
= 2
\][/tex]
This is correct.
So, the correct answer is option D.
