https://file.aiquickdraw.com/mathbot/file/1730482973564ocupnryy.jpg

The expression in the image is:

\[
4 \sqrt{12} \times 7 \sqrt{6}
\]

Let's simplify this step by step.

### Step 1: Break down the square roots

1. **Simplify \(\sqrt{12}\):**
   \[
   \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2 \sqrt{3}
   \]

2. **Simplify \(\sqrt{6}\):**
   \(\sqrt{6}\) is already in its simplest form, so we leave it as \(\sqrt{6}\).

Now, substitute these simplified values back into the expression:

\[
4 \sqrt{12} \times 7 \sqrt{6} = 4 \times 2 \sqrt{3} \times 7 \sqrt{6}
\]

### Step 2: Multiply the constants

Combine \(4 \times 2 \times 7\):

\[
4 \times 2 = 8
\]
\[
8 \times 7 = 56
\]

So we now have:

\[
56 \sqrt{3} \times \sqrt{6}
\]

### Step 3: Simplify the square roots

Combine \(\sqrt{3} \times \sqrt{6}\):

\[
\sqrt{3} \times \sqrt{6} = \sqrt{18} = \sqrt{9 \times 2} = 3 \sqrt{2}
\]

Now, substitute this back:

\[
56 \sqrt{3} \times \sqrt{6} = 56 \times 3 \sqrt{2} = 168 \sqrt{2}
\]

### Final Answer

\[
4 \sqrt{12} \times 7 \sqrt{6} = 168 \sqrt{2}
\]

Would you like a deeper explanation on any step? Here are some related questions:

1. How do we simplify square roots with composite numbers?
2. Why is \(\sqrt{12} = 2 \sqrt{3}\) considered a simplification?
3. Can this expression be further simplified in any context?
4. What if the original expression was addition instead of multiplication?
5. How would you handle an expression with higher roots, like cube roots?

**Tip:** When multiplying terms with square roots, try to simplify each root separately before combining them to make the process easier.

Learn how to simplify the radical expression 4√12 × 7√6. This solution demonstrates techniques for breaking down and multiplying radicals, including simplifying each radical before combining. Ideal for students in Grades 9-10.

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