https://file.aiquickdraw.com/mathbot/file/17258160314224tph2s3y.jpg

The expression shown in the image is:

\[
\frac{12}{3\sqrt{2} - 2}
\]

To simplify this, we can rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(3\sqrt{2} - 2\) is \(3\sqrt{2} + 2\).

### Steps:
1. Multiply both the numerator and the denominator by \(3\sqrt{2} + 2\):
\[
\frac{12}{3\sqrt{2} - 2} \times \frac{3\sqrt{2} + 2}{3\sqrt{2} + 2}
\]

2. Now, in the denominator, apply the difference of squares formula:
\[
(3\sqrt{2} - 2)(3\sqrt{2} + 2) = (3\sqrt{2})^2 - 2^2 = 18 - 4 = 14
\]

3. In the numerator, distribute \(12\) to \(3\sqrt{2} + 2\):
\[
12(3\sqrt{2} + 2) = 36\sqrt{2} + 24
\]

Thus, the simplified expression becomes:
\[
\frac{36\sqrt{2} + 24}{14}
\]

4. Finally, you can simplify this by dividing both terms in the numerator by 14:
\[
\frac{36\sqrt{2}}{14} + \frac{24}{14} = \frac{18\sqrt{2}}{7} + \frac{12}{7}
\]

Therefore, the simplified expression is:

\[
\frac{18\sqrt{2}}{7} + \frac{12}{7}
\]

Let me know if you'd like more details or explanations on this process!

Here are five questions related to this problem:
1. What is the conjugate of a binomial expression with a square root?
2. Why is rationalizing the denominator useful in this context?
3. How is the difference of squares formula applied in simplifying fractions?
4. Can this simplified form be expressed in decimal form, and if so, how?
5. What other techniques can be used to simplify expressions involving square roots?

**Tip**: When rationalizing denominators, always check for the simplest form by using the conjugate of the expression containing the square root.

Simplify the expression 12 / (3√2 - 2).

Learn how to simplify the expression 12 / (3√2 - 2) using algebraic techniques like rationalizing the denominator and applying the difference of squares formula. This problem is suitable for high school students in grades 9-12.

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