\left(\sqrt{7+\sqrt{3}}-\sqrt{7-\sqrt{3}}\right)

To simplify the expression 

\[
\sqrt{7+\sqrt{3}} - \sqrt{7-\sqrt{3}},
\]

we can try to rationalize it. Let’s rewrite it in a form that allows us to eliminate the square roots by multiplying by the conjugate:

\[
\left(\sqrt{7+\sqrt{3}} - \sqrt{7-\sqrt{3}}\right) \cdot \frac{\sqrt{7+\sqrt{3}} + \sqrt{7-\sqrt{3}}}{\sqrt{7+\sqrt{3}} + \sqrt{7-\sqrt{3}}}.
\]

This is equal to

\[
\frac{\left(\sqrt{7+\sqrt{3}} - \sqrt{7-\sqrt{3}}\right) \cdot \left(\sqrt{7+\sqrt{3}} + \sqrt{7-\sqrt{3}}\right)}{\sqrt{7+\sqrt{3}} + \sqrt{7-\sqrt{3}}}.
\]

In the numerator, we use the difference of squares formula, \((a - b)(a + b) = a^2 - b^2\):

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

Simplifying the terms in the numerator:

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

This fraction simplifies the expression but does not fully simplify it further without additional information or assumptions. If you’d like more details or further steps, let me know.

---

Would you like further breakdown of any steps?

1. How to approach similar radical expressions.
2. What the rationalization process involves.
3. How conjugates help in simplifying expressions.
4. Alternative methods to simplify complex square roots.
5. Why the expression does not fully simplify in some cases.

**Tip:** Using conjugates is especially useful for expressions with sums or differences involving square roots.

Understand how to simplify the radical expression \sqrt{7+\sqrt{3}} - \sqrt{7-\sqrt{3}} using algebraic methods like the difference of squares and conjugate rationalization. This problem covers concepts in radical expressions and algebraic simplification, suitable for Grades 10-12.

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