To solve the expression \( 7 : (2\sqrt{5} + 6) \), it is important to clarify whether the ":" symbol represents division. Assuming it does, the expression is equivalent to:

\[
\frac{7}{2\sqrt{5} + 6}
\]

To simplify this expression, one typically rationalizes the denominator:

1. Multiply the numerator and denominator by the conjugate of the denominator \(2\sqrt{5} - 6\):
   \[
   \frac{7}{2\sqrt{5} + 6} \times \frac{2\sqrt{5} - 6}{2\sqrt{5} - 6}
   \]

2. In the denominator, use the difference of squares:
   \[
   (2\sqrt{5})^2 - (6)^2 = 4 \times 5 - 36 = 20 - 36 = -16
   \]
   
3. In the numerator, distribute the 7:
   \[
   7 \times (2\sqrt{5} - 6) = 14\sqrt{5} - 42
   \]
   
4. The resulting expression is:
   \[
   \frac{14\sqrt{5} - 42}{-16} = -\frac{14\sqrt{5}}{16} + \frac{42}{16}
   \]
   
5. Simplify the fractions:
   \[
   -\frac{7\sqrt{5}}{8} + \frac{21}{8} = \frac{21 - 7\sqrt{5}}{8}
   \]

So, the simplified result is:

\[
\frac{21 - 7\sqrt{5}}{8}
\]

Would you like any additional details or have any questions?

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Here are five related questions to explore further:

1. How do you rationalize a denominator that contains a square root?
2. What is the conjugate of a binomial expression?
3. Can you simplify the expression \(\frac{5\sqrt{3}}{4\sqrt{2}}\)?
4. What is the difference of squares and how is it used in rationalization?
5. How do you add or subtract like terms involving square roots?

**Tip:** When rationalizing denominators, always multiply by the conjugate to eliminate square roots in the denominator.

Learn how to solve the expression 7 divided by (2 times the square root of 5 plus 6) step-by-step with detailed explanations. This problem involves rationalizing the denominator and simplifying algebraic expressions. Suitable for advanced high school students and beyond.

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