https://r2.erweima.ai/mathbot/image/1726575889137s127ogt6.jpg

The expression in the image appears to be:

\[
\frac{8}{3 - \sqrt{7}}
\]

To simplify this, we need to rationalize the denominator, which involves multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(3 - \sqrt{7}\) is \(3 + \sqrt{7}\).

### Step-by-step solution:

1. Multiply both the numerator and denominator by \(3 + \sqrt{7}\):

\[
\frac{8}{3 - \sqrt{7}} \times \frac{3 + \sqrt{7}}{3 + \sqrt{7}} = \frac{8(3 + \sqrt{7})}{(3 - \sqrt{7})(3 + \sqrt{7})}
\]

2. Simplify the denominator using the difference of squares formula:

\[
(3 - \sqrt{7})(3 + \sqrt{7}) = 3^2 - (\sqrt{7})^2 = 9 - 7 = 2
\]

3. Now, simplify the numerator:

\[
8(3 + \sqrt{7}) = 8 \cdot 3 + 8 \cdot \sqrt{7} = 24 + 8\sqrt{7}
\]

4. Therefore, the expression becomes:

\[
\frac{24 + 8\sqrt{7}}{2}
\]

5. Simplify further by dividing each term in the numerator by 2:

\[
\frac{24}{2} + \frac{8\sqrt{7}}{2} = 12 + 4\sqrt{7}
\]

### Final Answer:
\[
12 + 4\sqrt{7}
\]

Would you like further details on any step, or do you have any questions?

Here are five related questions:
1. How do you find the conjugate of a binomial with a square root?
2. Why is rationalizing the denominator useful in simplification?
3. Can you apply the same method to rationalize a denominator of \( \frac{5}{2 - \sqrt{3}} \)?
4. What is the difference of squares formula and how is it applied in simplifications like this?
5. How would you simplify \( \frac{7}{4 - \sqrt{5}} \) by rationalizing the denominator?

**Tip:** Rationalizing the denominator helps eliminate irrational numbers from the denominator, making the expression easier to handle, especially for further calculations.

Simplify the expression 8 / (3 - sqrt(7))

Learn how to simplify the expression 8 / (3 - sqrt(7)) using the rationalization method. This solution involves key algebraic concepts such as radicals and the difference of squares formula, making it suitable for students in Grades 9-11.

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