To find the expression equivalent to [tex]\(\frac{\sqrt[7]{x^2}}{\sqrt[5]{y^3}}\)[/tex], let's convert each part of the expression into an exponent form.
1. [tex]\(\sqrt[7]{x^2}\)[/tex] can be written as [tex]\(x^{\frac{2}{7}}\)[/tex] using the property of exponents that [tex]\(\sqrt[n]{x^m} = x^{\frac{m}{n}}\)[/tex].
2. Similarly, [tex]\(\sqrt[5]{y^3}\)[/tex] can be written as [tex]\(y^{\frac{3}{5}}\)[/tex].
So, the original expression [tex]\(\frac{\sqrt[7]{x^2}}{\sqrt[5]{y^3}}\)[/tex] can be rewritten as [tex]\(\frac{x^{\frac{2}{7}}}{y^{\frac{3}{5}}}\)[/tex].
Using the property of exponents [tex]\(\frac{a^m}{b^n} = a^m \cdot b^{-n}\)[/tex], we can further simplify this expression to:
[tex]\[ x^{\frac{2}{7}} \cdot y^{-\frac{3}{5}} \][/tex]
Therefore, the correct equivalent expression is:
[tex]\[ \boxed{\left( x^{\frac{2}{7}}\right) \left( y^{-\frac{3}{5}} \right)} \][/tex]
