The problem in the image is asking to simplify the radical expression:

$\sqrt{54y^8}$

Given that $y$ represents a positive real number, let's break this down step by step:

Step 1: Split the expression inside the square root

The expression under the square root is $54y^8$, which can be rewritten as:

$\sqrt{54y^8} = \sqrt{54} \times \sqrt{y^8}$

Step 2: Simplify the square root of $y^8$

Since the exponent of $y$ is 8, and the square root is the same as raising to the power of $\frac{1}{2}$, we can simplify $\sqrt{y^8}$ as:

$\sqrt{y^8} = y^{8/2} = y^4$

Step 3: Simplify the square root of 54

The square root of 54 can be simplified by factoring 54:

$54 = 9 \times 6 \quad \Rightarrow \quad \sqrt{54} = \sqrt{9 \times 6} = \sqrt{9} \times \sqrt{6} = 3\sqrt{6}$

Step 4: Combine the results

Now, combine the simplified parts:

$\sqrt{54y^8} = 3\sqrt{6} \times y^4 = 3y^4\sqrt{6}$

Thus, the simplified expression is:

$3y^4\sqrt{6}$

Would you like more details on this process or further clarification?

Here are some follow-up questions you might consider:

1. How do you handle radical expressions with odd exponents?
2. Can you generalize this process for any even exponent under the square root?
3. How would you handle simplifying $\sqrt{54y^9}$?
4. What are the steps for simplifying cube roots with even exponents?
5. How do you verify that your simplified answer is correct?

Tip: When simplifying radicals, always check for perfect square factors inside the radical to simplify further!