To simplify the expression \(\frac{4}{\sqrt{2} - 2\sqrt{3}}\), we need to rationalize the denominator. This involves getting rid of the square roots in the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator.
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].
