Given:
The pair of expressions in the options.
To find:
The pairs which show a pair of equivalent expressions.
Solution:
Property of exponent:
[tex](\sqrt[n]{x})^m=x^{\frac{m}{n}}[/tex]
In option A,
[tex](\sqrt[4]{81})^5=81^{\frac{5}{4}}[/tex]
So, option A is correct.
In option B,
[tex]6^{\frac{7}{2}}=(\sqrt[2]{6})^7[/tex]
[tex]6^{\frac{7}{2}}=(\sqrt{6})^7[/tex]
So, option B is correct.
In option C,
[tex]5^{\frac{2}{3}}=(\sqrt[3]{5})^2[/tex]
[tex]5^{\frac{2}{3}}\neq (\sqrt{5})^3[/tex]
So, option C is incorrect.
In option D,
[tex]7^{\frac{5}{7}}=(\sqrt[7]{7})^5[/tex]
[tex]7^{\frac{5}{7}}\neq (\sqrt{7})^5[/tex]
So, option D is incorrect.
