The problem asks to evaluate the following expressions: i. $36^{\frac{1}{2}}$ ii. $625^{\frac{1}{4}}$ iii. $10000^{\frac{1}{4}}$ iv. $49^{-\frac{3}{2}}$ v. $(\frac{27}{125})^{-\frac{2}{3}}$

1. Problem Description

The problem asks to evaluate the following expressions:

i. $36^{\frac{1}{2}}$

ii. $625^{\frac{1}{4}}$

iii. $10000^{\frac{1}{4}}$

iv. $49^{-\frac{3}{2}}$

v. $(\frac{27}{125})^{-\frac{2}{3}}$

2. Solution Steps

i. $36^{\frac{1}{2}} = \sqrt{36} = 6$

ii. $625^{\frac{1}{4}} = \sqrt[4]{625} = \sqrt[4]{5^4} = 5$

iii. $10000^{\frac{1}{4}} = \sqrt[4]{10000} = \sqrt[4]{10^4} = 10$

iv. $49^{-\frac{3}{2}} = (49^{\frac{1}{2}})^{-3} = (\sqrt{49})^{-3} = 7^{-3} = \frac{1}{7^3} = \frac{1}{343}$

v. $(\frac{27}{125})^{-\frac{2}{3}} = (\frac{125}{27})^{\frac{2}{3}} = ((\frac{125}{27})^{\frac{1}{3}})^2 = (\frac{\sqrt[3]{125}}{\sqrt[3]{27}})^2 = (\frac{5}{3})^2 = \frac{25}{9}$

3. Final Answer

i. 6

ii. 5

iii. 10

iv. $\frac{1}{343}$

v. $\frac{25}{9}$