https://file.aiquickdraw.com/mathbot/file/17274029154964ambz05p.jpg

Let's go through each option to find the **incorrect** one.

### Option a:
\[
(3 + \sqrt{2})(1 + \sqrt{2}) = 3 + 4\sqrt{2}
\]
First, expand the left-hand side:
\[
(3 + \sqrt{2})(1 + \sqrt{2}) = 3 \cdot 1 + 3 \cdot \sqrt{2} + \sqrt{2} \cdot 1 + \sqrt{2} \cdot \sqrt{2}
\]
This becomes:
\[
= 3 + 3\sqrt{2} + \sqrt{2} + 2 = 5 + 4\sqrt{2}
\]
Since the right-hand side is \(3 + 4\sqrt{2}\), this is **incorrect**.

### Option b:
\[
\frac{5}{3\sqrt{5}} = \frac{5(3 + \sqrt{5})}{4}
\]
The left-hand side can be simplified:
\[
\frac{5}{3\sqrt{5}} = \frac{5}{\sqrt{5}} \times \frac{1}{3} = \frac{5\sqrt{5}}{15} = \frac{\sqrt{5}}{3}
\]
Compare that to the right-hand side:
\[
\frac{5(3 + \sqrt{5})}{4}
\]
These are not equivalent, so this option is also **incorrect**.

### Option c:
\[
\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}
\]
Multiply the numerator and denominator on the left-hand side by \(\sqrt{2}\):
\[
\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{\sqrt{2}}{2}
\]
So this option is **correct**.

### Option d:
\[
3\sqrt{2} - \sqrt{8} = \sqrt{2}
\]
Simplify \(\sqrt{8}\):
\[
\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}
\]
So the left-hand side becomes:
\[
3\sqrt{2} - 2\sqrt{2} = \sqrt{2}
\]
This is **correct**.

### Conclusion:
- **a** is the incorrect answer as it does not simplify correctly.

Do you want any further clarifications or explanations? Here are some follow-up questions to consider:

1. How do you expand binomials like in option a?
2. What steps are involved in rationalizing a denominator as seen in option b?
3. Can you explain why \(\frac{1}{\sqrt{2}}\) is equivalent to \(\frac{\sqrt{2}}{2}\)?
4. Why does simplifying radicals help in verifying expressions like option d?
5. How do you know when two expressions are equal?

**Tip:** When simplifying, always check for common factors and use identities like \(\sqrt{a^2} = a\).

Choose the INCORRECT answer among the four options involving algebraic simplifications and square roots.

This algebra problem asks you to find the incorrect simplification involving binomial expansions, square roots, and rationalization. Explore step-by-step solutions to similar problems, suitable for Grades 10-12.

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