Option D: [tex]x\sqrt[9]{y^2}[/tex] is the expression equivalent to [tex]xy^{\frac{2}{9}}[/tex]
Explanation:
Option A: [tex]\sqrt{xy^9}[/tex]
The expression can be written as [tex]({xy^9})^{\frac{1}{2}[/tex]
Applying exponent rule, we get,
[tex]x^{\frac{1}{2}} y^{\frac{9}{2}}[/tex]
Thus, the expression [tex]\sqrt{xy^9}[/tex] is not equivalent to the expression [tex]xy^{\frac{2}{9}}[/tex]
Hence, Option A is not the correct answer.
Option B: [tex]\sqrt[9]{xy^2}[/tex]
The expression can be written as [tex]({xy^2})^{\frac{1}{9}[/tex]
Applying exponent rule, we get,
[tex]x^{\frac{1}{9}} y^{\frac{2}{9}}[/tex]
Thus, the expression [tex]\sqrt[9]{xy^2}[/tex] is not equivalent to the expression [tex]xy^{\frac{2}{9}}[/tex]
Hence, Option B is not the correct answer.
Option C: [tex]x\sqrt{y^9}[/tex]
The expression can be written as [tex]x(y^9)^{\frac{1}{2} }[/tex]
Applying exponent rule, we get,
[tex]x y^{\frac{9}{2}}[/tex]
Thus, the expression [tex]x\sqrt{y^9}[/tex] is not equivalent to the expression [tex]xy^{\frac{2}{9}}[/tex]
Hence, Option C is not the correct answer.
Option D: [tex]x\sqrt[9]{y^{2} }[/tex]
The expression can be written as [tex]x(y^2)^{\frac{1}{9} }[/tex]
Applying exponent rule, we get,
[tex]xy^{\frac{2}{9}}[/tex]
Thus, the expression [tex]xy^{\frac{2}{9}}[/tex] is equivalent to the expression [tex]xy^{\frac{2}{9}}[/tex]
Hence, Option D is the correct answer.
