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The given expression is:

\[
\frac{1 - \frac{1}{\sqrt{2}}}{1 + \frac{1}{\sqrt{2}}}
\]

We need to simplify this expression in the form \( a \pm b\sqrt{c} \), where \( a \), \( b \), and \( c \) are real numbers.

### Step 1: Rationalize the denominator
Multiply both the numerator and the denominator by the conjugate of the denominator:

\[
\text{Conjugate of } \left(1 + \frac{1}{\sqrt{2}}\right) \text{ is } \left(1 - \frac{1}{\sqrt{2}}\right).
\]

Now, multiply the expression:

\[
\frac{1 - \frac{1}{\sqrt{2}}}{1 + \frac{1}{\sqrt{2}}} \times \frac{1 - \frac{1}{\sqrt{2}}}{1 - \frac{1}{\sqrt{2}}} = \frac{\left(1 - \frac{1}{\sqrt{2}}\right)^2}{\left(1 + \frac{1}{\sqrt{2}}\right)\left(1 - \frac{1}{\sqrt{2}}\right)}.
\]

### Step 2: Simplify the denominator
The denominator is in the form \( (a + b)(a - b) = a^2 - b^2 \). So:

\[
(1 + \frac{1}{\sqrt{2}})(1 - \frac{1}{\sqrt{2}}) = 1^2 - \left(\frac{1}{\sqrt{2}}\right)^2 = 1 - \frac{1}{2} = \frac{1}{2}.
\]

### Step 3: Expand the numerator
The numerator is \( \left(1 - \frac{1}{\sqrt{2}}\right)^2 \):

\[
\left(1 - \frac{1}{\sqrt{2}}\right)^2 = 1^2 - 2 \cdot 1 \cdot \frac{1}{\sqrt{2}} + \left(\frac{1}{\sqrt{2}}\right)^2 = 1 - \frac{2}{\sqrt{2}} + \frac{1}{2}.
\]

Simplifying further:

\[
1 - \frac{2}{\sqrt{2}} + \frac{1}{2} = 1 - \sqrt{2} + \frac{1}{2}.
\]

### Step 4: Combine numerator and denominator
Now, combine the numerator and denominator:

\[
\frac{1 - \sqrt{2} + \frac{1}{2}}{\frac{1}{2}} = 2 \left( 1 - \sqrt{2} + \frac{1}{2} \right) = 2 - 2\sqrt{2} + 1 = 3 - 2\sqrt{2}.
\]

### Final Answer:
The expression simplifies to \( 3 - 2\sqrt{2} \).

Thus, in the form \( a - b\sqrt{c} \), we have:
- \( a = 3 \),
- \( b = 2 \),
- \( c = 2 \).

Do you have any questions or need further clarification? Here are some related problems to explore:

1. Simplify \( \frac{1 + \frac{1}{\sqrt{3}}}{1 - \frac{1}{\sqrt{3}}} \) in the form \( a \pm b\sqrt{c} \).
2. Find the value of \( \frac{1 + \frac{2}{\sqrt{5}}}{1 - \frac{2}{\sqrt{5}}} \).
3. Rationalize and simplify \( \frac{2 - \sqrt{3}}{2 + \sqrt{3}} \).
4. Expand and simplify \( (1 - \frac{1}{\sqrt{3}})^2 \).
5. Calculate \( \left(1 - \frac{2}{\sqrt{5}}\right)^2 \) and express the result in the form \( a \pm b\sqrt{c} \).

**Tip:** When rationalizing expressions with surds, always multiply by the conjugate of the denominator to simplify efficiently!

Simplify the following in the form of a surd: (1 - 1/√2) / (1 + 1/√2). Answer in the form a ± b√c.

Learn how to simplify (1 - 1/√2) / (1 + 1/√2) using surds and rationalizing the denominator. This problem explains key algebraic techniques and covers concepts of rationalization. Suitable for students in Grades 9-12.

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