To determine which expression is equivalent to [tex]\( y^{\frac{2}{5}} \)[/tex], we need to use the properties of exponents and radicals.
1. Understanding Exponential Expressions:
- The given expression is [tex]\( y^{\frac{2}{5}} \)[/tex].
2. Rewriting the Exponent:
- We know that an expression in the form of [tex]\( a^{\frac{m}{n}} \)[/tex] can be rewritten using radicals. Specifically, [tex]\( a^{\frac{m}{n}} \)[/tex] is the same as the [tex]\( n \)[/tex]-th root of [tex]\( a^m \)[/tex]. Mathematically, this is written as:
[tex]\[
a^{\frac{m}{n}} = \sqrt[n]{a^m}
\][/tex]
3. Applying this to [tex]\( y^{\frac{2}{5}} \)[/tex]:
- Using the property above, let’s rewrite [tex]\( y^{\frac{2}{5}} \)[/tex]:
[tex]\[
y^{\frac{2}{5}} = \sqrt[5]{y^2}
\][/tex]
4. Comparing with Given Options:
- Let’s compare [tex]\( \sqrt[5]{y^2} \)[/tex] with the available options:
A. [tex]\( \sqrt[5]{2y} \)[/tex] indicates the fifth root of [tex]\( 2y \)[/tex], which is not equivalent to [tex]\( y^{\frac{2}{5}} \)[/tex].
B. [tex]\( \sqrt[5]{y^2} \)[/tex] is exactly the same as [tex]\( y^{\frac{2}{5}} \)[/tex].
C. [tex]\( \sqrt{y^5} \)[/tex] indicates the square root of [tex]\( y^5 \)[/tex], which is not equivalent to [tex]\( y^{\frac{2}{5}} \)[/tex].
D. [tex]\( 2 \sqrt[5]{y} \)[/tex] indicates 2 times the fifth root of [tex]\( y \)[/tex], which is not equivalent to [tex]\( y^{\frac{2}{5}} \)[/tex].
Given these comparisons, the correct answer is:
B. [tex]\( \sqrt[5]{y^2} \)[/tex]
