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To simplify the expression \(\frac{7 - \sqrt{5}}{2 + \sqrt{5}}\) in the form \(a + b\sqrt{5}\), where \(a\) and \(b\) are integers, we can proceed as follows:

### Step-by-Step Working:

1. **Rationalize the denominator**:
   To eliminate the square root in the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(2 + \sqrt{5}\) is \(2 - \sqrt{5}\).

   So, we multiply:
   \[
   \frac{7 - \sqrt{5}}{2 + \sqrt{5}} \times \frac{2 - \sqrt{5}}{2 - \sqrt{5}} = \frac{(7 - \sqrt{5})(2 - \sqrt{5})}{(2 + \sqrt{5})(2 - \sqrt{5})}
   \]

2. **Simplify the denominator**:
   The denominator is a difference of squares:
   \[
   (2 + \sqrt{5})(2 - \sqrt{5}) = 2^2 - (\sqrt{5})^2 = 4 - 5 = -1
   \]

3. **Expand the numerator**:
   Now expand \((7 - \sqrt{5})(2 - \sqrt{5})\) using distribution:
   \[
   7(2 - \sqrt{5}) - \sqrt{5}(2 - \sqrt{5}) = 7 \times 2 - 7 \times \sqrt{5} - \sqrt{5} \times 2 + \sqrt{5} \times \sqrt{5}
   \]
   This simplifies to:
   \[
   14 - 7\sqrt{5} - 2\sqrt{5} + 5 = 19 - 9\sqrt{5}
   \]

4. **Combine terms**:
   So the numerator becomes:
   \[
   19 - 9\sqrt{5}
   \]

5. **Final expression**:
   Now divide the numerator by the denominator:
   \[
   \frac{19 - 9\sqrt{5}}{-1} = -19 + 9\sqrt{5}
   \]

Thus, the simplified expression is:
\[
-19 + 9\sqrt{5}
\]

So, in the form \(a + b\sqrt{5}\), we have:
\[
a = -19 \quad \text{and} \quad b = 9
\]

Would you like more details on any part of this process?

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Here are 5 related questions for further practice:

1. Simplify \(\frac{3 + \sqrt{7}}{1 - \sqrt{7}}\) in the form \(a + b\sqrt{7}\).
2. Simplify \(\frac{5 - \sqrt{2}}{3 + \sqrt{2}}\).
3. What is the conjugate of \(4 + \sqrt{3}\), and why is it useful in simplifications?
4. Simplify \(\frac{1 + \sqrt{3}}{1 - \sqrt{3}}\).
5. Express \(\frac{4 + 2\sqrt{6}}{3 - \sqrt{6}}\) in the form \(a + b\sqrt{6}\).

**Tip**: Whenever you encounter a denominator with a square root, multiplying by the conjugate is a powerful technique to simplify expressions.

Simplify (7 - √5) / (2 + √5), giving your answer in the form a + b√5, where a and b are integers. Show clear working out.

Learn how to simplify the algebraic expression (7 - √5) / (2 + √5) using rationalization and the difference of squares theorem. This problem focuses on simplifying surds and rationalizing denominators. Suitable for students in Grades 10-12.

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