To determine which of the given options is equivalent to [tex]\( 6^{2/5} \)[/tex], let's analyze the expression step-by-step.
1. Understanding the expression [tex]\( 6^{2/5} \)[/tex]:
- The expression [tex]\( 6^{2/5} \)[/tex] can be interpreted as either:
- The 5th root of [tex]\( 6 \)[/tex] raised to the power of 2, i.e., [tex]\( (\sqrt[5]{6})^2 \)[/tex], or
- [tex]\( 6 \)[/tex] raised to the power of [tex]\( 2 \)[/tex] and then taking the 5th root of the result.
2. Rewrite the options to match [tex]\( 6^{2/5} \)[/tex]:
- Option a) [tex]\( (\sqrt[6]{5})^2 \)[/tex]:
- This means the 6th root of [tex]\( 5 \)[/tex] raised to the power of 2.
- This can be written as [tex]\( (5^{1/6})^2 = 5^{2/6} = 5^{1/3} \)[/tex].
- Clearly, this doesn't match [tex]\( 6^{2/5} \)[/tex].
- Option b) [tex]\( (\sqrt[6]{2})^5 \)[/tex]:
- This means the 6th root of [tex]\( 2 \)[/tex] raised to the power of 5.
- This can be written as [tex]\( (2^{1/6})^5 = 2^{5/6} \)[/tex].
- Clearly, this doesn't match [tex]\( 6^{2/5} \)[/tex].
- Option c) [tex]\( (\sqrt[5]{6})^2 \)[/tex]:
- This means the 5th root of [tex]\( 6 \)[/tex] raised to the power of 2.
- This can be written as [tex]\( (6^{1/5})^2 = 6^{2/5} \)[/tex].
- This matches our original expression [tex]\( 6^{2/5} \)[/tex].
- Option d) [tex]\( (\sqrt[5]{2})^6 \)[/tex]:
- This means the 5th root of [tex]\( 2 \)[/tex] raised to the power of 6.
- This can be written as [tex]\( (2^{1/5})^6 = 2^{6/5} \)[/tex].
- Clearly, this doesn't match [tex]\( 6^{2/5} \)[/tex].
3. Conclusion:
- After analyzing all options, we see that option c), [tex]\( (\sqrt[5]{6})^2 \)[/tex], matches our original expression [tex]\( 6^{2/5} \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{c} \][/tex]
