To simplify $(8x^2y)^{\frac{1}{3}}$, follow these steps:

Step 1: Distribute the exponent $\frac{1}{3}$ to each term inside the parentheses.

$(8x^2y)^{\frac{1}{3}} = 8^{\frac{1}{3}} \cdot (x^2)^{\frac{1}{3}} \cdot y^{\frac{1}{3}}$

Step 2: Simplify each part.

1. $8^{\frac{1}{3}}$ is the cube root of 8, which is 2. $8^{\frac{1}{3}} = 2$

2. $(x^2)^{\frac{1}{3}}$ simplifies by multiplying the exponents. $(x^2)^{\frac{1}{3}} = x^{\frac{2}{3}}$

3. $y^{\frac{1}{3}}$ remains as is.

Step 3: Combine the results.

$2 \cdot x^{\frac{2}{3}} \cdot y^{\frac{1}{3}}$

So, $(8x^2y)^{\frac{1}{3}} = 2x^{\frac{2}{3}}y^{\frac{1}{3}}$.

Would you like further details or have any questions?

Here are 5 related questions to explore:

1. How would you simplify $(27x^3)^{\frac{1}{3}}$?
2. What happens if the exponent inside the parentheses is negative, like $(x^{-2}y)^{\frac{1}{3}}$?
3. Can you apply the same rules for fractional exponents in expressions with roots?
4. What are some common mistakes to avoid when simplifying expressions with fractional exponents?
5. How do you handle simplifying expressions with multiple variables inside parentheses?

Tip: Remember, fractional exponents represent roots. For example, $a^{1/n}$ means the $n$-th root of $a$.