Let's simplify the given expression step by step. The expression is:

\[
7\sqrt{8} - 5\sqrt{18} + \sqrt{27}\sqrt{8} - 5\sqrt{18} + \sqrt{2}
\]

### Step 1: Simplify the Square Roots
1. **Simplify \(\sqrt{8}\):**
   \[
   \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}
   \]
   So, \(7\sqrt{8} = 7 \times 2\sqrt{2} = 14\sqrt{2}\).

2. **Simplify \(\sqrt{18}\):**
   \[
   \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}
   \]
   So, \(-5\sqrt{18} = -5 \times 3\sqrt{2} = -15\sqrt{2}\).

3. **Simplify \(\sqrt{27}\):**
   \[
   \sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}
   \]

4. **Simplify \(\sqrt{27}\sqrt{8}\):**
   \(\sqrt{27}\) and \(\sqrt{8}\) simplified earlier become \(3\sqrt{3}\) and \(2\sqrt{2}\), respectively. So:
   \[
   \sqrt{27}\sqrt{8} = 3\sqrt{3} \times 2\sqrt{2} = 6\sqrt{6}
   \]

5. **Simplify \(\sqrt{2}\):**
   \(\sqrt{2}\) is already in its simplest form.

### Step 2: Substitute and Combine Like Terms
Now, substitute the simplified terms back into the expression:
\[
14\sqrt{2} - 15\sqrt{2} + 6\sqrt{6} - 15\sqrt{2} + \sqrt{2}
\]

Combine like terms:
- \(14\sqrt{2} - 15\sqrt{2} - 15\sqrt{2} + \sqrt{2} = -15\sqrt{2} + \sqrt{2} = -14\sqrt{2}\)
- The remaining term is \(6\sqrt{6}\).

So, the final simplified expression is:
\[
-14\sqrt{2} + 6\sqrt{6}
\]

This is the simplified form of the expression.

Would you like more details or have any questions? Here are five related questions for practice:

1. Simplify \(5\sqrt{12} + 3\sqrt{75} - 2\sqrt{48}\).
2. Factor and simplify \(2\sqrt{50} + \sqrt{18}\).
3. How do you simplify the expression \(\sqrt{72} + \sqrt{50} - \sqrt{18}\)?
4. What is the simplified form of \(3\sqrt{8} + 7\sqrt{32}\)?
5. Solve and simplify \(2\sqrt{3} \times 3\sqrt{6}\).

**Tip:** Always simplify square roots by factoring out perfect squares.

Learn how to simplify the expression 7√8 - 5√18 + √27√8 - 5√18 + √2 step-by-step. This problem involves simplifying square roots and combining like terms. Suitable for high school students studying algebra and radical expressions.

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