To determine which expression is equivalent to [tex]\( y^{\frac{2}{8}} \)[/tex], let's simplify the given expression step by step.
1. Simplify the Exponent:
The expression given is [tex]\( y^{\frac{2}{8}} \)[/tex]. We can simplify the fraction [tex]\(\frac{2}{8}\)[/tex] as follows:
[tex]\[
\frac{2}{8} = \frac{1}{4}
\][/tex]
Therefore, [tex]\( y^{\frac{2}{8}} \)[/tex] simplifies to:
[tex]\[
y^{\frac{1}{4}}
\][/tex]
2. Rewrite in Radical Form:
The expression [tex]\( y^{\frac{1}{4}} \)[/tex] can be rewritten using radical notation. The exponent [tex]\(\frac{1}{4}\)[/tex] indicates the fourth root of [tex]\(y\)[/tex]:
[tex]\[
y^{\frac{1}{4}} = \sqrt[4]{y}
\][/tex]
Now, we need to compare this simplified expression, [tex]\( \sqrt[4]{y} \)[/tex], with the given options:
- Option A: [tex]\( \sqrt[5]{2y} \)[/tex]:
This represents the fifth root of [tex]\(2y\)[/tex], which is not equivalent to [tex]\( \sqrt[4]{y} \)[/tex].
- Option B: [tex]\( \sqrt[5]{y^2} \)[/tex]:
This represents the fifth root of [tex]\( y^2 \)[/tex], which is not equivalent to [tex]\( \sqrt[4]{y} \)[/tex].
- Option C: [tex]\( \sqrt{y^5} \)[/tex]:
This represents the square root of [tex]\( y^5 \)[/tex], which is not equivalent to [tex]\( \sqrt[4]{y} \)[/tex].
- Option D: [tex]\( 2 \sqrt[6]{y} \)[/tex]:
This represents 2 multiplied by the sixth root of [tex]\( y \)[/tex], which is not equivalent to [tex]\( \sqrt[4]{y} \)[/tex].
None of the options provided match the simplified form [tex]\( \sqrt[4]{y} \)[/tex]. Therefore, if we strictly adhere to the given set of options, none of them is correct. However, based on the correct simplification process, [tex]\( y^{\frac{2}{8}} \)[/tex] is indeed equivalent to [tex]\( \sqrt[4]{y} \)[/tex].
Hence, there seems to be a discrepancy in the provided choices, as the correct answer should be:
[tex]\[ \sqrt[4]{y} \][/tex]
