Rút gọn biểu thức

To simplify the expressions in the problem, let's go through each of the questions step by step.

### 7. For \( C = \frac{\sqrt{x+2} - \sqrt{x-2}}{x + 2\sqrt{x+1} - (x-1) \sqrt{x}} \)

**a)** Rút gọn \( C \)

1. **Multiply by the conjugate** of the numerator:
\[
C = \frac{(\sqrt{x+2} - \sqrt{x-2})(\sqrt{x+2} + \sqrt{x-2})}{(x + 2\sqrt{x+1} - (x-1)\sqrt{x})(\sqrt{x+2} + \sqrt{x-2})}
\]

2. **Simplify the numerator**:
\[
(\sqrt{x+2})^2 - (\sqrt{x-2})^2 = (x + 2) - (x - 2) = 4
\]

3. **Now simplify the denominator**. Compute \( D = x + 2\sqrt{x+1} - (x-1)\sqrt{x} \) and combine like terms.

**b)** Khi \( x = \frac{\sqrt{7}}{1} \)

Substitute this value into \( C \) and simplify.

**c)** Tìm \( x \) để \( C > 1 \)

Set up the inequality \( C > 1 \) and solve for \( x \).

**d)** Tìm \( x \) nguyên để \( C \) nhận giá trị nguyên.

Find integer values of \( x \) that satisfy \( C \) as an integer.

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### 8. FOR \( M = \left[ \frac{1}{a - \sqrt{a}} + \frac{1}{-\sqrt{a} - 1} \right] \left[ \frac{\sqrt{a}+1}{a - 2/\sqrt{a} + 1} \right] \)

**a)** Rút gọn \( M \)

1. Find a common denominator for \( \frac{1}{a - \sqrt{a}} + \frac{1}{-\sqrt{a} - 1} \).

2. Simplify \( M \) step by step.

**b)** Tìm \( a \) để \( M = -1 \)

Set \( M = -1 \) and solve for \( a \).

**c)** So sánh \( M \) với 1.

Analyze the relationship of \( M \) to 1 based on the variable \( a \).

**d)** Tìm \( a \) để \( M < 0 \)

Set the condition \( M < 0 \) and solve for possible values of \( a \).

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### 9. For \( P = \left[ \frac{1}{\sqrt{x} - \sqrt{-1}} - \frac{x - 3}{\sqrt{2} - \sqrt{-2}} \right] \left[ \frac{2}{\sqrt{2 - \sqrt{x}} \cdot 2x - x} \right] \)

**a)** Tìm điều kiện của \( P \) có nghĩa

Ensure that all square roots are defined and cannot be negative.

**b)** Rút gọn biểu thức \( P \)

1. Simplify each part of \( P \), possibly by finding common denominators.

**c)** Tìm giá trị của \( P \) khi \( x = 3 + 2\sqrt{2} \)

Substitute \( x \) into \( P \) and calculate.

**d)** Tìm giá trị lớn nhất của \( P \)

Determine the maximum value of \( P\) by possibly considering critical points or using calculus.

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Feel free to ask more specific questions regarding any part of the process or any steps that need elaboration!