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When you encounter a fraction with a square root in the denominator, like \( \frac{5}{9+\sqrt{7}} \), your goal is to make the denominator rational. Rationalizing the denominator is the process of eliminating the square roots or irrational numbers from the bottom part of a fraction without changing the value of the expression.

To rationalize the denominator, you multiply both the numerator and the denominator by a term that will remove the square root when multiplied by the denominator. This term often involves the 'conjugate' of the binomial in the denominator, which leads us directly into our next concept.

The conjugate of a binomial, such as \( 9 + \sqrt{7} \), is formed by changing the sign between the two terms. In this case, the conjugate is \( 9 - \sqrt{7} \).

Why is this important? When you multiply a binomial by its conjugate, the result is a 'difference of squares,' which simplifies nicely. To rationalize our original expression's denominator, we multiply both the numerator and the denominator by \( 9 - \sqrt{7} \). This effectively eliminates the square root in the denominator through multiplication using this conjugate.

Multiplying by the conjugate is a crucial technique in algebra that simplifies expressions into a more usable form, especially when dealing with square roots.

This brings us to our final related concept, the difference of squares formula, which states that \( (a+b)(a-b) = a^2 - b^2 \).

When we multiply the conjugate of a binomial by the original binomial, we are essentially using this formula. It works because the cross-terms cancel out, leaving only the squared terms. For \( (9+\sqrt{7})(9-\sqrt{7}) \), following this formula, our denominator simplifies to \( 9^2 - (\sqrt{7})^2 \), which equals \( 81 - 7 \).

Understanding the difference of squares formula is vital in algebra, as it frequently appears in various problems, such as factoring or simplifying expressions. It helps transform complex expres\sions with minimal effort, leading to the final simplified expression of \( \frac{45 - 5\sqrt{7}}{74} \) in our original problem.