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].
