Let's go through each of the problems quickly using the rule given:
$a^{\frac{1}{n}} = \sqrt[n]{a}$

Solution Steps:

1. Problem 1: $4^{\frac{1}{2}}$

   • This expression represents the square root of 4.
   • $\sqrt{4} = 2$
   • Answer: $4^{\frac{1}{2}} = 2$

2. Problem 2: $8^{\frac{1}{3}}$

   • This expression represents the cube root of 8.
   • $\sqrt[3]{8} = 2$
   • Answer: $8^{\frac{1}{3}} = 2$

3. Problem 3: $(-27)^{\frac{1}{3}}$

   • This expression represents the cube root of -27.
   • $\sqrt[3]{-27} = -3$
   • Answer: $(-27)^{\frac{1}{3}} = -3$

These solutions follow the definition given in the instructions. Let me know if you'd like additional details or have any questions.

Related Questions:

1. How can we simplify expressions with exponents greater than 1, like $a^{\frac{m}{n}}$?
2. What are the rules