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## Sagot :

1.

**Identify the conjugate of the denominator**:

The conjugate of \(\sqrt{2} - 2\sqrt{3}\) is \(\sqrt{2} + 2\sqrt{3}\).

2.

**Multiply the numerator and the denominator by the conjugate**:

[tex]\[ \frac{4}{\sqrt{2} - 2\sqrt{3}} \times \frac{\sqrt{2} + 2\sqrt{3}}{\sqrt{2} + 2\sqrt{3}} \][/tex]

3.

**Distribute the numerator**:

[tex]\[ 4 \cdot (\sqrt{2} + 2\sqrt{3}) = 4\sqrt{2} + 8\sqrt{3} \][/tex]

4.

**Use the difference of squares formula for the denominator**:

[tex]\[ (\sqrt{2} - 2\sqrt{3})(\sqrt{2} + 2\sqrt{3}) = (\sqrt{2})^2 - (2\sqrt{3})^2 \][/tex]

[tex]\[ (\sqrt{2})^2 = 2 \][/tex]

[tex]\[ (2\sqrt{3})^2 = 4 \cdot 3 = 12 \][/tex]

[tex]\[ \text{So, the denominator becomes } 2 - 12 = -10 \][/tex]

5.

**Combine the results**:

[tex]\[ \frac{4\sqrt{2} + 8\sqrt{3}}{-10} \][/tex]

6.

**Simplify the fraction**:

[tex]\[ \frac{4\sqrt{2} + 8\sqrt{3}}{-10} = \frac{4(\sqrt{2} + 2\sqrt{3})}{-10} = \frac{4(\sqrt{2} + 2\sqrt{3})}{-10} = -\frac{2(\sqrt{2} + 2\sqrt{3})}{5} \][/tex]

Finally, if we compute the numerical value of this simplified version, we get:

[tex]\[ \sqrt{2} + 2\sqrt{3} \approx 3.9026521420086793 \][/tex]

Multiplying by -\(\frac{2}{5}\), the result is approximately:

[tex]\[ -\frac{2 \cdot 3.9026521420086793}{5} \approx -1.9513260710043396 \][/tex]

Thus, the simplified version of the expression [tex]\(\frac{4}{\sqrt{2} - 2\sqrt{3}}\)[/tex] is [tex]\(-\frac{2(\sqrt{2} + 2\sqrt{3})}{5}\)[/tex] and its approximate numerical value is [tex]\(-1.9513260710043396\)[/tex]. The calculated result confirms that the value of the denominator is [tex]\(-10\)[/tex] and the final numerical result is indeed approximately [tex]\(-1.9513260710043396\)[/tex].