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Sagot :
To solve the given problem, we need to simplify the expression [tex]\(\left(2^{\frac{1}{4}}\right)^4\)[/tex]. Let's go through each of the provided options and determine which one accurately simplifies the given statement.
### Option A:
[tex]\[ \left(2^{\frac{1}{4}}\right)^4=2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}}=2 \cdot \left(\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}\right)=2 \cdot \frac{4}{4}=2 \][/tex]
- Breaking down the expression multiplication [tex]\(2 \cdot \left(\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4}\right)\)[/tex]: This misinterprets the multiplication of exponents. The statement here incorrectly doubles [tex]\(2\)[/tex], not respecting the rules of exponentiation correctly.
### Option B:
[tex]\[ \left(2^{\frac{1}{4}}\right)^4=2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}}=2^{\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}}=2^{\frac{4}{4}}=2^1=2 \][/tex]
- Here, the product of the exponents is: [tex]\(2^{\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4}} = 2^{1} = 2\)[/tex]. This is the correct way to handle multiplication of like bases with exponents.
### 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 uses multiplication of the exponents instead of adding them. The correct operation should be [tex]\(2^{\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4}} = 2^{1}\)[/tex].
### Option D:
[tex]\[ \left(2^{\frac{1}{4}}\right)^4=2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}}=4 \cdot 2^{\frac{1}{4}}=4 \cdot \frac{1}{4} \cdot 2=2 \][/tex]
- This option incorrectly multiplies the result by 4 and introduces the mixup with the exponents. The simplification [tex]\(4 \cdot 2^{\frac{1}{4}}\)[/tex] and then [tex]\((2 \cdot \frac{1}{4}) = 2\)[/tex] is incorrect.
### Conclusion
After reviewing all the options, Option B correctly simplifies the expression:
[tex]\[ \left(2^{\frac{1}{4}}\right)^4=2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}}=2^{\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4}}=2^{1}=2 \][/tex]
Thus, the correct simplification is given in Option B.
### Option A:
[tex]\[ \left(2^{\frac{1}{4}}\right)^4=2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}}=2 \cdot \left(\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}\right)=2 \cdot \frac{4}{4}=2 \][/tex]
- Breaking down the expression multiplication [tex]\(2 \cdot \left(\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4}\right)\)[/tex]: This misinterprets the multiplication of exponents. The statement here incorrectly doubles [tex]\(2\)[/tex], not respecting the rules of exponentiation correctly.
### Option B:
[tex]\[ \left(2^{\frac{1}{4}}\right)^4=2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}}=2^{\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}}=2^{\frac{4}{4}}=2^1=2 \][/tex]
- Here, the product of the exponents is: [tex]\(2^{\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4}} = 2^{1} = 2\)[/tex]. This is the correct way to handle multiplication of like bases with exponents.
### 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 uses multiplication of the exponents instead of adding them. The correct operation should be [tex]\(2^{\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4}} = 2^{1}\)[/tex].
### Option D:
[tex]\[ \left(2^{\frac{1}{4}}\right)^4=2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}}=4 \cdot 2^{\frac{1}{4}}=4 \cdot \frac{1}{4} \cdot 2=2 \][/tex]
- This option incorrectly multiplies the result by 4 and introduces the mixup with the exponents. The simplification [tex]\(4 \cdot 2^{\frac{1}{4}}\)[/tex] and then [tex]\((2 \cdot \frac{1}{4}) = 2\)[/tex] is incorrect.
### Conclusion
After reviewing all the options, Option B correctly simplifies the expression:
[tex]\[ \left(2^{\frac{1}{4}}\right)^4=2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}}=2^{\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4}}=2^{1}=2 \][/tex]
Thus, the correct simplification is given in Option B.
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