https://file.aiquickdraw.com/mathbot/file/1729489331381vyaxd7as.jpg

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!

Simplify √(54y^8), assuming y represents a positive real number.

Learn how to simplify the radical expression √(54y^8), step by step. This example includes concepts such as radical simplification and exponent rules. Suitable for high school students in Grades 9-12.

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