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The expression provided is:

\[
\frac{\sqrt{28} - 8}{\sqrt{7} + 2}
\]
and we are asked to show that it can be written as \( 10 - 4\sqrt{7} \).

### Step-by-Step Solution:

1. **Simplify \(\sqrt{28}\):**
   \[
   \sqrt{28} = \sqrt{4 \times 7} = \sqrt{4} \times \sqrt{7} = 2\sqrt{7}
   \]
   Now, the expression becomes:
   \[
   \frac{2\sqrt{7} - 8}{\sqrt{7} + 2}
   \]

2. **Rationalize the denominator:**
   To rationalize the denominator, multiply both the numerator and the denominator by the conjugate of the denominator, which is \(\sqrt{7} - 2\):
   \[
   \frac{(2\sqrt{7} - 8)(\sqrt{7} - 2)}{(\sqrt{7} + 2)(\sqrt{7} - 2)}
   \]

3. **Simplify the denominator:**
   The denominator is a difference of squares:
   \[
   (\sqrt{7} + 2)(\sqrt{7} - 2) = \sqrt{7}^2 - 2^2 = 7 - 4 = 3
   \]
   So the expression becomes:
   \[
   \frac{(2\sqrt{7} - 8)(\sqrt{7} - 2)}{3}
   \]

4. **Expand the numerator:**
   Now expand the numerator:
   \[
   (2\sqrt{7} - 8)(\sqrt{7} - 2) = 2\sqrt{7} \times \sqrt{7} + 2\sqrt{7} \times (-2) - 8 \times \sqrt{7} - 8 \times (-2)
   \]
   Simplifying each term:
   \[
   2\sqrt{7} \times \sqrt{7} = 2 \times 7 = 14
   \]
   \[
   2\sqrt{7} \times (-2) = -4\sqrt{7}
   \]
   \[
   -8 \times \sqrt{7} = -8\sqrt{7}
   \]
   \[
   -8 \times (-2) = 16
   \]
   Putting it all together:
   \[
   14 - 4\sqrt{7} - 8\sqrt{7} + 16 = 14 + 16 - (4\sqrt{7} + 8\sqrt{7})
   \]
   Simplify:
   \[
   30 - 12\sqrt{7}
   \]

5. **Divide by 3:**
   Now divide the expression by 3:
   \[
   \frac{30 - 12\sqrt{7}}{3} = \frac{30}{3} - \frac{12\sqrt{7}}{3} = 10 - 4\sqrt{7}
   \]

Thus, the expression simplifies to \( 10 - 4\sqrt{7} \), as required.

---

Would you like more details or have any questions?

Here are five related questions you might want to explore:
1. How do you find the conjugate of a binomial with a square root?
2. Why do we rationalize the denominator in expressions involving square roots?
3. How does the difference of squares formula work in this context?
4. Can you explain the importance of simplifying radicals before proceeding?
5. How would you handle this expression if there were cube roots instead?

**Tip:** When rationalizing denominators, always look for the conjugate of the denominator to simplify square root terms effectively.

Show that (sqrt(28) - 8) / (sqrt(7) + 2) can be written as 10 - 4sqrt(7).

Learn how to simplify and rationalize the expression (sqrt(28) - 8) / (sqrt(7) + 2) by using the difference of squares and radical simplification techniques. This step-by-step solution is suitable for students in Grades 10-12.

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