Answer :

Sure, let's simplify the given expression step-by-step:

Given expression:
[tex]\[ \frac{\sqrt{6}}{\sqrt{2} + \sqrt{3}} \][/tex]

To rationalize the denominator, we can multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of [tex]\(\sqrt{2} + \sqrt{3}\)[/tex] is [tex]\(\sqrt{2} - \sqrt{3}\)[/tex].

Step-by-step solution:

1. Multiply the expression by the conjugate of the denominator:
[tex]\[ \frac{\sqrt{6}}{\sqrt{2} + \sqrt{3}} \times \frac{\sqrt{2} - \sqrt{3}}{\sqrt{2} - \sqrt{3}} \][/tex]
This gives us:
[tex]\[ \frac{\sqrt{6} \cdot (\sqrt{2} - \sqrt{3})}{(\sqrt{2} + \sqrt{3})(\sqrt{2} - \sqrt{3})} \][/tex]

2. Simplify the denominator using the difference of squares formula [tex]\((a^2 - b^2) = (a - b)(a + b)\)[/tex]:
[tex]\[ (\sqrt{2})^2 - (\sqrt{3})^2 = 2 - 3 = -1 \][/tex]

3. Simplify the numerator:
[tex]\[ \sqrt{6} \cdot (\sqrt{2} - \sqrt{3}) \][/tex]
This results in:
[tex]\[ \sqrt{6} \cdot \sqrt{2} - \sqrt{6} \cdot \sqrt{3} \][/tex]
Simplifying further:
[tex]\[ \sqrt{12} - \sqrt{18} \][/tex]
[tex]\[ \sqrt{4 \times 3} - \sqrt{9 \times 2} \][/tex]
[tex]\[ 2\sqrt{3} - 3\sqrt{2} \][/tex]
So the numerator is:
[tex]\[ 2\sqrt{3} - 3\sqrt{2} \approx -0.77853907198153 \][/tex]

4. Combine the numerator and denominator:
[tex]\[ \frac{2\sqrt{3} - 3\sqrt{2}}{-1} \][/tex]

5. Finalize the simplification:
[tex]\[ - (2\sqrt{3} - 3\sqrt{2}) \][/tex]
This results in:
[tex]\[ 3\sqrt{2} - 2\sqrt{3} \approx 0.77853907198153 \][/tex]

So, the simplified form of the expression [tex]\(\frac{\sqrt{6}}{\sqrt{2} + \sqrt{3}}\)[/tex] is approximately [tex]\(0.77853907198153\)[/tex].