To determine which given expression is equivalent to [tex]\( xy^{\frac{2}{9}} \)[/tex], let's analyze each option step-by-step and simplify where necessary.
### Analyzing the Expressions:
1. [tex]\(\sqrt{xy^9}\)[/tex]:
- This expression can be written as [tex]\((xy^9)^{\frac{1}{2}}\)[/tex].
- Using the power of a product property: [tex]\((ab)^c = a^c b^c\)[/tex], we get [tex]\(x^{\frac{1}{2}} \cdot (y^9)^{\frac{1}{2}}\)[/tex].
- Simplify further: [tex]\(x^{\frac{1}{2}} \cdot y^{\frac{9}{2}}\)[/tex].
- Comparing this with [tex]\( xy^{\frac{2}{9}} \)[/tex], it is evident they are not equivalent since the exponents of [tex]\(y\)[/tex] are different.
2. [tex]\(\sqrt[9]{xy^2}\)[/tex]:
- This expression can be written as [tex]\((xy^2)^{\frac{1}{9}}\)[/tex].
- Using the power of a product property again: [tex]\( (ab)^c = a^c b^c \)[/tex], we get [tex]\( x^{\frac{1}{9}} \cdot (y^2)^{\frac{1}{9}}\)[/tex].
- Simplify further: [tex]\(x^{\frac{1}{9}} \cdot y^{\frac{2}{9}}\)[/tex].
- Comparing this with [tex]\(xy^{\frac{2}{9}}\)[/tex], it is evident they are not equivalent since [tex]\(x\)[/tex] has a different exponent.
3. [tex]\( x\left(\sqrt{y^9}\right) \)[/tex]:
- This expression can be written as [tex]\( x(y^9)^{\frac{1}{2}} \)[/tex].
- Simplify the inner expression: [tex]\( x \cdot y^{\frac{9}{2}} \)[/tex].
- Comparing this with [tex]\( xy^{\frac{2}{9}} \)[/tex], it is evident they are not equivalent because the exponents of [tex]\(y\)[/tex] are different.
4. [tex]\( x\left(\sqrt[9]{y^2}\right) \)[/tex]:
- This expression can be written as [tex]\( x(y^2)^{\frac{1}{9}} \)[/tex].
- Simplify the inner expression: [tex]\( x \cdot y^{\frac{2}{9}} \)[/tex].
- Comparing this with [tex]\( xy^{\frac{2}{9}} \)[/tex], we see that they match perfectly.
### Conclusion:
The correct equivalent expression to [tex]\( xy^{\frac{2}{9}} \)[/tex] is:
[tex]\[ \boxed{x\left(\sqrt[9]{y^2}\right)} \][/tex]
