Answer :
To determine which expression is equivalent to \( y^{\frac{2}{5}} \), we need to simplify and analyze each given option one by one. Let’s work through each option.
### Option A: \(\sqrt[5]{2 y}\)
This expression represents the fifth root of \(2y\). Written in fractional exponent form, it is:
[tex]\[ \sqrt[5]{2 y} = (2 y)^{\frac{1}{5}} \][/tex]
This is not equivalent to \( y^{\frac{2}{5}} \), because the exponent of \( y \) is \(\frac{1}{5}\) rather than \(\frac{2}{5}\).
### Option B: \(\sqrt[5]{y^2}\)
This expression represents the fifth root of \( y^2 \). Written in fractional exponent form, it is:
[tex]\[ \sqrt[5]{y^2} = \left(y^2\right)^{\frac{1}{5}} \][/tex]
Using the property of exponents \( (a^m)^n = a^{m \cdot n} \):
[tex]\[ \left(y^2\right)^{\frac{1}{5}} = y^{2 \cdot \frac{1}{5}} = y^{\frac{2}{5}} \][/tex]
This matches the original expression \( y^{\frac{2}{5}} \).
### Option C: \(\sqrt{y^5}\)
This expression represents the square root of \( y^5 \). Written in fractional exponent form, it is:
[tex]\[ \sqrt{y^5} = \left(y^5\right)^{\frac{1}{2}} \][/tex]
Using the property of exponents:
[tex]\[ \left(y^5\right)^{\frac{1}{2}} = y^{5 \cdot \frac{1}{2}} = y^{\frac{5}{2}} \][/tex]
This is not equivalent to \( y^{\frac{2}{5}} \), because the exponent of \( y \) is \(\frac{5}{2}\) rather than \(\frac{2}{5}\).
### Option D: \(2 \sqrt[5]{y}\)
This expression represents 2 times the fifth root of \( y \). Written in fractional exponent form, it is:
[tex]\[ 2 \sqrt[5]{y} = 2 \cdot y^{\frac{1}{5}} \][/tex]
This form retains the 2 as a constant multiplier, and the exponent of \( y \) is \(\frac{1}{5}\), not \(\frac{2}{5}\).
### Conclusion
From the analysis, the expression that is equivalent to \( y^{\frac{2}{5}} \) is:
[tex]\[ \boxed{\text{B.} \ \sqrt[5]{y^2}} \][/tex]
### Option A: \(\sqrt[5]{2 y}\)
This expression represents the fifth root of \(2y\). Written in fractional exponent form, it is:
[tex]\[ \sqrt[5]{2 y} = (2 y)^{\frac{1}{5}} \][/tex]
This is not equivalent to \( y^{\frac{2}{5}} \), because the exponent of \( y \) is \(\frac{1}{5}\) rather than \(\frac{2}{5}\).
### Option B: \(\sqrt[5]{y^2}\)
This expression represents the fifth root of \( y^2 \). Written in fractional exponent form, it is:
[tex]\[ \sqrt[5]{y^2} = \left(y^2\right)^{\frac{1}{5}} \][/tex]
Using the property of exponents \( (a^m)^n = a^{m \cdot n} \):
[tex]\[ \left(y^2\right)^{\frac{1}{5}} = y^{2 \cdot \frac{1}{5}} = y^{\frac{2}{5}} \][/tex]
This matches the original expression \( y^{\frac{2}{5}} \).
### Option C: \(\sqrt{y^5}\)
This expression represents the square root of \( y^5 \). Written in fractional exponent form, it is:
[tex]\[ \sqrt{y^5} = \left(y^5\right)^{\frac{1}{2}} \][/tex]
Using the property of exponents:
[tex]\[ \left(y^5\right)^{\frac{1}{2}} = y^{5 \cdot \frac{1}{2}} = y^{\frac{5}{2}} \][/tex]
This is not equivalent to \( y^{\frac{2}{5}} \), because the exponent of \( y \) is \(\frac{5}{2}\) rather than \(\frac{2}{5}\).
### Option D: \(2 \sqrt[5]{y}\)
This expression represents 2 times the fifth root of \( y \). Written in fractional exponent form, it is:
[tex]\[ 2 \sqrt[5]{y} = 2 \cdot y^{\frac{1}{5}} \][/tex]
This form retains the 2 as a constant multiplier, and the exponent of \( y \) is \(\frac{1}{5}\), not \(\frac{2}{5}\).
### Conclusion
From the analysis, the expression that is equivalent to \( y^{\frac{2}{5}} \) is:
[tex]\[ \boxed{\text{B.} \ \sqrt[5]{y^2}} \][/tex]
