https://file.aiquickdraw.com/mathbot/file/17291948441716wu96f8s.jpg

The expression given in the image is:

\[
8w^2 \sqrt{8w x^2} - x \sqrt{50w^5}
\]

Let's break this down and simplify it step by step.

### Step 1: Simplify the first square root term, \( \sqrt{8w x^2} \)

- \( 8w x^2 \) can be broken down into \( 4 \times 2 \times w \times x^2 \).
- The square root of \( 4x^2 \) is \( 2x \), leaving \( \sqrt{2w} \).

So, the first term becomes:

\[
8w^2 \cdot 2x \cdot \sqrt{2w} = 16w^2 x \sqrt{2w}
\]

### Step 2: Simplify the second square root term, \( \sqrt{50w^5} \)

- \( 50w^5 \) can be written as \( 25 \times 2 \times w^4 \times w \).
- The square root of \( 25w^4 \) is \( 5w^2 \), leaving \( \sqrt{2w} \).

So, the second term becomes:

\[
x \cdot 5w^2 \cdot \sqrt{2w} = 5xw^2 \sqrt{2w}
\]

### Step 3: Combine the simplified terms

Now we have:

\[
16w^2 x \sqrt{2w} - 5xw^2 \sqrt{2w}
\]

Since both terms share the common factor \( w^2 x \sqrt{2w} \), we can factor it out:

\[
(16 - 5)w^2 x \sqrt{2w} = 11w^2 x \sqrt{2w}
\]

### Final Simplified Expression:
\[
11w^2 x \sqrt{2w}
\]

Would you like more details on any of these steps? Here are some related questions to explore:

1. How do you simplify expressions with square roots involving variables?
2. What properties of square roots and exponents help in this simplification?
3. How does factoring common terms work when variables and radicals are involved?
4. What happens if the radical contains higher powers of variables?
5. How can you check if a simplified expression is fully reduced?

**Tip:** Always check for common factors across terms before simplifying complex expressions!

Simplify as much as possible: 8w^2 √(8w x^2) - x √(50w^5)

Learn how to simplify the radical expression 8w^2 √(8w x^2) - x √(50w^5) by factoring and simplifying square roots. This problem involves key concepts such as the properties of square roots and exponents, and is suitable for Grades 10-12.

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