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Which expression is equivalent to [tex]\(\frac{\sqrt[7]{x^2}}{\sqrt[5]{y^3}}\)[/tex]? Assume [tex]\(y \neq 0\)[/tex].

A. [tex]\(\left(x^{\frac{2}{7}}\right)\left(y^{-\frac{3}{5}}\right)\)[/tex]

B. [tex]\(\binom{\frac{2}{7}}{x^7}\binom{\frac{5}{3}}{x^3}\)[/tex]

C. [tex]\(\left(x^{\frac{2}{7}}\right)\left(y^{\frac{3}{5}}\right)\)[/tex]

D. [tex]\(\left(\frac{7}{2}\right)\left(y^{\frac{5}{3}}\right)\)[/tex]

Sagot :

GinnyAnswer
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]
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