To simplify the expression \( \frac{1}{3} + \frac{2}{a^{\frac{2}{3}}} \), we need to determine if any of the provided options are equivalent to it.
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Option A: \( \frac{a^{\frac{2}{9}} + 2}{a^{\frac{2}{3}}} \)
This expression does not match the structure we have since the term \( a^{\frac{2}{9}} \) would not arise from the original expression.
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Option B: \( \frac{2}{a^{\frac{1}{3}}} \)
Again, this one does not match, and it doesn’t incorporate the \( \frac{1}{3} \) term correctly.
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Option C: \( \frac{a^{\frac{1}{3}} + 2}{a^{\frac{2}{3}}} \)
Here, we can combine terms under a common denominator: \[ \frac{a^{\frac{1}{3}}}{a^{\frac{2}{3}}} + \frac{2}{a^{\frac{2}{3}}} = a^{\frac{1}{3} - \frac{2}{3}} + \frac{2}{a^{\frac{2}{3}}} = a^{-\frac{1}{3}} + \frac{2}{a^{\frac{2}{3}}} \] This is not equivalent to the original expression.
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Option D: \( \frac{a + 2}{a^{\frac{2}{3}}} \)
Similar reasoning as above applies and would lead us to incorrect forms.
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Option E: \( \frac{a^{\frac{2}{3}} + 2}{a} \)
To compare this with our original expression appears not to yield a direct equivalent transformation either.
Conclusion: The original expression is \( \frac{1}{3} + \frac{2}{a^{\frac{2}{3}}} \). The upcoming answer does not match perfectly with any of the suggested options explicitly.
After checking the calculations and comparisons, none of the options is equivalent to the expression \( \frac{1}{3} + \frac{2}{a^{\frac{2}{3}}} \). Thus, it seems there might be an error in the provided answer choices or misunderstanding of simplification methods needed.
1 answer