To rationalize the expression \( \frac{1}{\sqrt{7} - \sqrt{2} + \sqrt{3}} \), we need to remove the square roots from the denominator.

The first step in rationalizing this expression is to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate is formed by changing the signs of the square roots in the denominator. Since the denominator involves three terms, \( \sqrt{7} - \sqrt{2} + \sqrt{3} \), the conjugate would be \( \sqrt{7} + \sqrt{2} - \sqrt{3} \).

Let's proceed with the rationalization:

\[
\frac{1}{\sqrt{7} - \sqrt{2} + \sqrt{3}} \times \frac{\sqrt{7} + \sqrt{2} - \sqrt{3}}{\sqrt{7} + \sqrt{2} - \sqrt{3}}
\]

This results in:

\[
\frac{\sqrt{7} + \sqrt{2} - \sqrt{3}}{(\sqrt{7} - \sqrt{2} + \sqrt{3})(\sqrt{7} + \sqrt{2} - \sqrt{3})}
\]

Now, let's simplify the denominator using the difference of squares formula:

\[
(\sqrt{7} - \sqrt{2} + \sqrt{3})(\sqrt{7} + \sqrt{2} - \sqrt{3}) = (\sqrt{7})^2 - (\sqrt{2} - \sqrt{3})^2
\]

First, compute the squares:

\[
(\sqrt{7})^2 = 7
\]

\[
(\sqrt{2} - \sqrt{3})^2 = (\sqrt{2})^2 - 2(\sqrt{2})(\sqrt{3}) + (\sqrt{3})^2 = 2 - 2\sqrt{6} + 3 = 5 - 2\sqrt{6}
\]

So the denominator becomes:

\[
7 - (5 - 2\sqrt{6}) = 7 - 5 + 2\sqrt{6} = 2 + 2\sqrt{6}
\]

Thus, the rationalized expression is:

\[
\frac{\sqrt{7} + \sqrt{2} - \sqrt{3}}{2 + 2\sqrt{6}}
\]

To simplify further, you can factor out the 2 from the denominator:

\[
\frac{\sqrt{7} + \sqrt{2} - \sqrt{3}}{2(1 + \sqrt{6})}
\]

Finally, the rationalized expression is:

\[
\frac{\sqrt{7} + \sqrt{2} - \sqrt{3}}{2(1 + \sqrt{6})}
\]

Would you like more details or have any specific questions about this process? Here are a few related questions you might find interesting:

1. How does the conjugate help in rationalizing denominators with square roots?
2. What is the difference of squares formula, and how does it simplify expressions?
3. Can this technique be applied to expressions with more than two terms in the denominator?
4. What happens if the terms in the denominator are more complex, like involving cube roots?
5. How do you handle irrational expressions in the numerator when rationalizing?

**Tip**: Always check if the denominator can be simplified after rationalizing; sometimes factoring out common terms can make the final result cleaner!

Learn how to rationalize the denominator of the expression 1/√7 - √2 + √3 step-by-step. This guide includes the use of conjugates and the difference of squares formula. Suitable for students in Grades 9-11.

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