To solve the problem of finding an expression equivalent to [tex]\(\frac{2}{9}\)[/tex], let's carefully analyze the given options:
Given four expressions:
1. [tex]\(\sqrt{x y^9}\)[/tex]
2. [tex]\(\sqrt[9]{x y^2}\)[/tex]
3. [tex]\(x\left(\sqrt{y^9}\right)\)[/tex]
4. [tex]\(x\left(\sqrt[9]{y^2}\right)\)[/tex]
We need to determine the one that simplifies to an expression involving the fractional exponent [tex]\(\frac{2}{9}\)[/tex].
### Evaluating Each Option
1. [tex]\(\sqrt{x y^9}\)[/tex]
- This can be written as [tex]\((x y^9)^{1/2}\)[/tex].
- Simplifying, we get [tex]\(x^{1/2} \cdot (y^9)^{1/2} = x^{1/2} \cdot y^{9/2}\)[/tex].
2. [tex]\(\sqrt[9]{x y^2}\)[/tex]
- This can be expressed as [tex]\((x y^2)^{1/9}\)[/tex].
- Simplifying, we get [tex]\(x^{1/9} \cdot (y^2)^{1/9} = x^{1/9} \cdot y^{2/9}\)[/tex].
3. [tex]\(x\left(\sqrt{y^9}\right)\)[/tex]
- Inside the parentheses, [tex]\(\sqrt{y^9} = (y^9)^{1/2} = y^{9/2}\)[/tex].
- Thus, the expression becomes [tex]\(x \cdot y^{9/2}\)[/tex].
4. [tex]\(x\left(\sqrt[9]{y^2}\right)\)[/tex]
- Inside the parentheses, [tex]\(\sqrt[9]{y^2} = (y^2)^{1/9} = y^{2/9}\)[/tex].
- Thus, the expression simplifies to [tex]\(x \cdot y^{2/9}\)[/tex].
### Matching with [tex]\(\frac{2}{9}\)[/tex]
We see that the expression [tex]\(y^{2/9}\)[/tex] appears in the fourth option. Therefore, the correct expression matching the required fractional exponent [tex]\(\frac{2}{9} \)[/tex] is:
[tex]\[ x\left(\sqrt[9]{y^2}\right) \][/tex]
Thus, the correct choice is the fourth option, [tex]\(x\left(\sqrt[9]{y^2}\right)\)[/tex].
The answer to this problem is:
4
