We can simplify the expression \( \sqrt{8} + S\sqrt{2} - 6\sqrt{2} \).

### Step 1: Simplify \( \sqrt{8} \)
We know that:
\[
\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}
\]

### Step 2: Combine Like Terms
Now, substitute \( \sqrt{8} \) with \( 2\sqrt{2} \):
\[
2\sqrt{2} + S\sqrt{2} - 6\sqrt{2}
\]

Combine the terms involving \( \sqrt{2} \):
\[
(2 + S - 6)\sqrt{2}
\]
\[
(S - 4)\sqrt{2}
\]

The simplified result is:
\[
(S - 4)\sqrt{2}
\]

This is the final simplified expression in terms of \( S \). If you have a specific value for \( S \), you can substitute it to get a numerical result.

Would you like further details or clarification on any part of this?

Here are five related questions:
1. What are the properties of square roots that can simplify expressions?
2. How do you combine like terms involving radicals?
3. How do you solve equations involving radicals?
4. Can \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \) be applied in all cases?
5. How do you rationalize a denominator with square roots?

**Tip:** Always simplify square roots by factoring under the radical sign to find perfect squares.

Learn how to simplify the radical expression √8 + S√2 - 6√2 by breaking down the steps: simplifying square roots and combining like terms. This example covers important algebraic concepts such as simplifying radicals and using the distributive property, suitable for students in Grades 9-11.

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