https://file.aiquickdraw.com/mathbot/file/1726303161447912lx5oa.jpg

The expression shown in the image is:

\[
6w^2 \left( \sqrt{80w} - \sqrt{20w^5} \right)
\]

We will now simplify this step by step.

### Step 1: Simplifying the square roots
- \(\sqrt{80w}\) can be simplified as:
  \[
  \sqrt{80w} = \sqrt{16 \times 5 \times w} = 4\sqrt{5w}
  \]

- \(\sqrt{20w^5}\) can be simplified as:
  \[
  \sqrt{20w^5} = \sqrt{4 \times 5 \times w^4 \times w} = 2w^2\sqrt{5w}
  \]

### Step 2: Substitute back into the expression
We now have:
\[
6w^2 \left( 4\sqrt{5w} - 2w^2\sqrt{5w} \right)
\]

### Step 3: Factor out common terms
Factor out the common term \(2\sqrt{5w}\):
\[
6w^2 \times 2\sqrt{5w} \left( 2 - w^2 \right)
\]

### Step 4: Simplify the expression
Multiply \(6w^2\) with \(2\sqrt{5w}\):
\[
12w^2\sqrt{5w} \left( 2 - w^2 \right)
\]

Thus, the simplified form of the expression is:
\[
12w^2\sqrt{5w} \left( 2 - w^2 \right)
\]

Would you like further clarification on any part of this process?

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Here are 5 related questions to expand on this:
1. How do you simplify square roots involving variables?
2. What is the process of factoring expressions like the one above?
3. Can you always factor out common terms in any expression?
4. How do exponents affect the process of simplifying square roots?
5. How can we apply this process to more complex algebraic expressions?

**Tip:** When simplifying square roots of expressions with variables, look for perfect squares and factor them out to make the process easier.

Simplify 6w^2 (√80w − √20w^5). Assume that the variable represents a positive real number.

Discover how to simplify the expression 6w^2 (√80w − √20w^5) with a step-by-step breakdown. This problem involves algebraic concepts such as simplifying radicals, factoring, and applying the properties of exponents. Ideal for students in Grades 9-11.

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