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Sagot :
Let's go through each of the given choices step-by-step to determine the accurate simplification for the statement:
[tex]\(\sqrt[4]{2} = 2^{\frac{1}{4}}\)[/tex].
### Choice 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]
- Explanation:
- Using the properties of exponents: [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex].
- Add the exponents: [tex]\(\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} = 1\)[/tex].
- Therefore, [tex]\(2^1 = 2\)[/tex].
This choice is correctly simplified and accurate.
### Choice B:
[tex]\[ \left(2^{\frac{1}{4}}\right)^4 = 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} = 4 \cdot 2^{\frac{1}{4}} = 4 \cdot \frac{1}{4} \cdot 2 = 2 \][/tex]
- Explanation:
- Initially incorrectly combines the [tex]\(2^{\frac{1}{4}}\)[/tex] terms as multiplication by 4.
- Incorrectly calculates [tex]\(4 \cdot \frac{1}{4} \cdot 2\)[/tex].
This choice contains errors in the operations and simplification, leading to an incorrect result.
### Choice C:
[tex]\[ \left(2^{\frac{1}{4}}\right)^4 = 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{6}} \cdot 2^{\frac{1}{4}} = 2^{\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4}} = 2^{\frac{4}{4}} = 2^1 = 2 \][/tex]
- Explanation:
- A misprint exists, where [tex]\(2^{\frac{1}{6}}\)[/tex] is included incorrectly.
- Even though the final result is simplified similarly, the step here has an incorrect intermediate exponent term ([tex]\(2^{\frac{1}{6}}\)[/tex]).
This choice has an incorrect element but gives the correct result at the end because of a mistake in the explanation.
### Choice D:
[tex]\[ \left(2^{\frac{1}{4}}\right)^4 = 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} = 2 \cdot \left(\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4}\right) = 2 \cdot \frac{4}{4} = 2 \][/tex]
- Explanation:
- Incorrectly separates the 2 which should not be applied in the context of the exponent addition.
This choice contains an incorrect mathematical setup for the simplification.
### Conclusion:
The correct choice, given the detailed analysis and explanation, is Choice A.
[tex]\(\sqrt[4]{2} = 2^{\frac{1}{4}}\)[/tex].
### Choice 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]
- Explanation:
- Using the properties of exponents: [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex].
- Add the exponents: [tex]\(\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} = 1\)[/tex].
- Therefore, [tex]\(2^1 = 2\)[/tex].
This choice is correctly simplified and accurate.
### Choice B:
[tex]\[ \left(2^{\frac{1}{4}}\right)^4 = 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} = 4 \cdot 2^{\frac{1}{4}} = 4 \cdot \frac{1}{4} \cdot 2 = 2 \][/tex]
- Explanation:
- Initially incorrectly combines the [tex]\(2^{\frac{1}{4}}\)[/tex] terms as multiplication by 4.
- Incorrectly calculates [tex]\(4 \cdot \frac{1}{4} \cdot 2\)[/tex].
This choice contains errors in the operations and simplification, leading to an incorrect result.
### Choice C:
[tex]\[ \left(2^{\frac{1}{4}}\right)^4 = 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{6}} \cdot 2^{\frac{1}{4}} = 2^{\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4}} = 2^{\frac{4}{4}} = 2^1 = 2 \][/tex]
- Explanation:
- A misprint exists, where [tex]\(2^{\frac{1}{6}}\)[/tex] is included incorrectly.
- Even though the final result is simplified similarly, the step here has an incorrect intermediate exponent term ([tex]\(2^{\frac{1}{6}}\)[/tex]).
This choice has an incorrect element but gives the correct result at the end because of a mistake in the explanation.
### Choice D:
[tex]\[ \left(2^{\frac{1}{4}}\right)^4 = 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} \cdot 2^{\frac{1}{4}} = 2 \cdot \left(\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4}\right) = 2 \cdot \frac{4}{4} = 2 \][/tex]
- Explanation:
- Incorrectly separates the 2 which should not be applied in the context of the exponent addition.
This choice contains an incorrect mathematical setup for the simplification.
### Conclusion:
The correct choice, given the detailed analysis and explanation, is Choice A.
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