perform the operation and simplify if possible (6square root 2 + square root 2 ( 5 square root 2 - 4 square root 3)

To perform the operation and simplify \(6\sqrt{2} + \sqrt{2} (5\sqrt{2} - 4\sqrt{3})\), follow these steps:

1. **Distribute \(\sqrt{2}\) inside the parentheses:**

\[
\sqrt{2} (5\sqrt{2} - 4\sqrt{3}) = \sqrt{2} \cdot 5\sqrt{2} - \sqrt{2} \cdot 4\sqrt{3}
\]

2. **Simplify each term:**

\[
\sqrt{2} \cdot 5\sqrt{2} = 5 (\sqrt{2} \cdot \sqrt{2}) = 5 \cdot 2 = 10
\]

\[
\sqrt{2} \cdot 4\sqrt{3} = 4 (\sqrt{2} \cdot \sqrt{3}) = 4\sqrt{6}
\]

So the expression becomes:

\[
6\sqrt{2} + 10 - 4\sqrt{6}
\]

3. **Combine like terms:**

There are no like terms to combine further since \(6\sqrt{2}\), \(10\), and \(4\sqrt{6}\) are all distinct. Thus, the simplified form is:

\[
6\sqrt{2} + 10 - 4\sqrt{6}
\]

This is the simplest form of the given expression.

If you have any questions or need further details, feel free to ask!

Here are 5 more questions you might consider asking:
1. How do you simplify expressions involving multiple radicals?
2. What is the process for rationalizing the denominator of a radical expression?
3. How do you add and subtract radicals with different indices?
4. Can you explain how to multiply and divide radical expressions?
5. What are some tips for recognizing like terms when working with radicals?

**Tip:** When working with radicals, always simplify each term completely before attempting to combine or perform operations.

Learn how to simplify the expression 6√2 + √2(5√2 - 4√3) step-by-step. This problem involves radicals and simplification techniques, suitable for advanced high school students.

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