We need to simplify the expression:

$\frac{1}{\sqrt{3} + \sqrt{7} + \sqrt{6}}$

Step 1: Multiply by the Conjugate

To rationalize the denominator, we use the conjugate. Since we have three terms, one approach is to multiply by a modified form of the conjugate to simplify radicals.

Let’s multiply the numerator and denominator by $\sqrt{3} + \sqrt{7} - \sqrt{6}$:

$\frac{1}{\sqrt{3} + \sqrt{7} + \sqrt{6}} \times \frac{\sqrt{3} + \sqrt{7} - \sqrt{6}}{\sqrt{3} + \sqrt{7} - \sqrt{6}}$

Step 2: Expand the Denominator

Using the identity $(a+b+c)(a+b-c) = a^2 + b^2 - c^2 - 2bc$, we compute:

$(\sqrt{3} + \sqrt{7} + \sqrt{6})(\sqrt{3} + \sqrt{7} - \sqrt{6})$

Expanding:

$(\sqrt{3} + \sqrt{7})^2 - (\sqrt{6})^2$

Computing each term:

$(\sqrt{3} + \sqrt{7})^2 = 3 + 7 + 2\sqrt{21} = 10 + 2\sqrt{21}$

$(\sqrt{6})^2 = 6$

So the denominator simplifies to:

$10 + 2\sqrt{21} - 6 = 4 + 2\sqrt{21}$

Step 3: Simplify

Now our expression is:

$\frac{\sqrt{3} + \sqrt{7} - \sqrt{6}}{4 + 2\sqrt{21}}$

Factor out 2 from the denominator:

$\frac{\sqrt{3} + \sqrt{7} - \sqrt{6}}{2(2 + \sqrt{21})}$

Split:

$\frac{1}{2} \times \frac{\sqrt{3} + \sqrt{7} - \sqrt{6}}{2 + \sqrt{21}}$

Step 4: Multiply by the Conjugate Again

Multiply numerator and denominator by $2 - \sqrt{21}$:

$\frac{(\sqrt{3} + \sqrt{7} - \sqrt{6})(2 - \sqrt{21})}{(2+\sqrt{21})(2-\sqrt{21})}$

Since $(2+\sqrt{21})(2-\sqrt{21}) = 4 - 21 = -17$, we now compute the numerator:

$(\sqrt{3} + \sqrt{7} - \sqrt{6})(2 - \sqrt{21})$

Expanding:

$2\sqrt{3} + 2\sqrt{7} - 2\sqrt{6} - \sqrt{63} - \sqrt{147} + \sqrt{126}$

Since $\sqrt{63} = 3\sqrt{7}$, $\sqrt{147} = 7\sqrt{3}$, and $\sqrt{126} = 3\sqrt{14}$, we get:

$2\sqrt{3} + 2\sqrt{7} - 2\sqrt{6} - 3\sqrt{7} - 7\sqrt{3} + 3\sqrt{14}$

$-5\sqrt{3} - \sqrt{7} - 2\sqrt{6} + 3\sqrt{14}$

Dividing by -17, we get:

$\frac{-5\sqrt{3} - \sqrt{7} - 2\sqrt{6} + 3\sqrt{14}}{-34} = \frac{5\sqrt{3} + \sqrt{7} + 2\sqrt{6} - 3\sqrt{14}}{34}$

This is the simplified form.

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Final Answer:

$\frac{5\sqrt{3} + \sqrt{7} + 2\sqrt{6} - 3\sqrt{14}}{34}$

Would you like a more detailed explanation for any step?

Here are some related problems:

1. Simplify $\frac{1}{\sqrt{5} + \sqrt{2} + \sqrt{3}}$
2. Rationalize $\frac{1}{\sqrt{7} - \sqrt{3}}$
3. Compute $\frac{1}{\sqrt{a} + \sqrt{b} + \sqrt{c}}$ using general steps
4. Evaluate $\frac{2}{\sqrt{8} + \sqrt{18}}$
5. Simplify $\frac{3}{\sqrt{5} + \sqrt{3}}$

Tip:

When rationalizing a denominator with three terms, try multiplying by a conjugate expression that cancels out some radicals systematically.