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Select the simplification that accurately explains the following statement: [tex]\sqrt[4]{2}=2^{\frac{1}{4}}[/tex]

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]

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} \cdot \frac{1}{4} \cdot \frac{1}{4} \cdot \frac{1}{4}}=2^{\frac{4}{4}}=2^1=2[/tex]

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}}=4 \cdot 2^{\frac{1}{4}}=4 \cdot \frac{1}{4} \cdot 2=2[/tex]

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]

Sagot :

GinnyAnsweridn
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.
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