Rút gọn các biểu thức sau: a) \(\frac{1}{2} \sqrt{48 - 2\sqrt{75}} - \frac{\sqrt{11}}{5} + \sqrt{\frac{3}{2}}\)

To simplify the expression in part a:

\[
\frac{1}{2} \sqrt{48 - 2\sqrt{75}} - \frac{\sqrt{11}}{5} + \sqrt{\frac{3}{2}}
\]

1. **Simplify \(\sqrt{75}\)**:
\[
\sqrt{75} = \sqrt{25 \cdot 3} = 5\sqrt{3}
\]

2. **Substituting into the expression**:
\[
48 - 2\sqrt{75} = 48 - 2(5\sqrt{3}) = 48 - 10\sqrt{3}
\]

3. **Now calculate \(\sqrt{48 - 10\sqrt{3}}\)**:

Let \(x = \sqrt{48 - 10\sqrt{3}}\). We assume it can be expressed as \(\sqrt{a} - \sqrt{b}\):
\[
x^2 = a + b - 2\sqrt{ab}
\]
Therefore,
\[
a + b = 48 \quad \text{and} \quad -2\sqrt{ab} = -10\sqrt{3} \implies \sqrt{ab} = 5\sqrt{3} \implies ab = 75
\]

Solving the system:
\(t^2 - 48t + 75 = 0\):
\[
t = \frac{48 \pm \sqrt{48^2 - 4 \cdot 75}}{2} = \frac{48 \pm \sqrt{2304 - 300}}{2} = \frac{48 \pm \sqrt{2004}}{2}
\]

Approximating \(\sqrt{2004} \approx 44.83\):
\[
t_1 \approx 46.415, \quad t_2 \approx 1.585
\]

Using the values \(a \approx 46.415\) and \(b \approx 1.585\), we get:
\[
\sqrt{48 - 10\sqrt{3}} = \sqrt{46.415} - \sqrt{1.585}
\]

4. **Putting it back in the expression**:
\[
\frac{1}{2} (\sqrt{a} - \sqrt{b}) - \frac{\sqrt{11}}{5} + \sqrt{\frac{3}{2}}
\]

5. **Calculating \(\sqrt{\frac{3}{2}}\)**:
\[
\sqrt{\frac{3}{2}} = \frac{\sqrt{6}}{2}
\]

Now replace and simplify as needed:
\[
\text{Final expression} = \frac{1}{2\sqrt{46.415}} - \frac{\sqrt{11}}{5} + \frac{\sqrt{6}}{2}
\]

This can be calculated further or approximated depending on your need.

For an exact result, you would leave it in this format as above. If you prefer numerical values, plug them into a calculator.

If you want the full simplification for all parts you've listed, just follow the methodology above for each of them. Let me know how else you would like to simplify or if you have any specific questions!