To solve the problem of finding which expression is equivalent to [tex]\(\frac{2}{9}\)[/tex], we need to evaluate each given algebraic expression to see whether it simplifies to [tex]\(\frac{2}{9}\)[/tex]. Let's evaluate them one by one:
1. Expression 1: [tex]\(\sqrt{x y^9}\)[/tex]
- This expression would need further specification of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values to evaluate, so let's hold off on this one as it doesn't lend itself to an immediate simplification to [tex]\(\frac{2}{9}\)[/tex].
2. Expression 2: [tex]\(\sqrt[9]{x y^2}\)[/tex]
- Similar to expression 1, it also requires a further specification of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values, so we'll hold off on this for the same reasons.
3. Expression 3: [tex]\(x \left( \sqrt{y^9} \right)\)[/tex]
- Once again, an immediate simplification isn't clear and it requires specifics on [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
4. Expression 4: [tex]\(x \left( \sqrt[9]{y^2} \right)\)[/tex]
- This simplifies to [tex]\(x \cdot y^{2/9}\)[/tex] which can potentially align with [tex]\(\frac{2}{9}\)[/tex] under typical algebraic simplifications if specific [tex]\(x\)[/tex] and [tex]\(y\)[/tex] relationships are entered.
Based on the structured examination, expression 4 more closely conforms to an algebraic simplification resulting in a form similar to [tex]\(\frac{2}{9}\)[/tex]. Thus, the expression equivalent to [tex]\(\frac{2}{9}\)[/tex] is:
[tex]\[ x \left( \sqrt[9]{y^2} \right) \][/tex]
Therefore, the correct expression among the given options is:
[tex]\[ \mathbf{ x \left( \sqrt[9]{y^2} \right) } \][/tex]
So, the correct index, considering it starts from zero, is:
3
