The expression to simplify is:

\[
\frac{8}{2 + \sqrt{5}}
\]

To simplify this expression, we can multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \(2 + \sqrt{5}\) is \(2 - \sqrt{5}\).

\[
\frac{8}{2 + \sqrt{5}} \times \frac{2 - \sqrt{5}}{2 - \sqrt{5}} = \frac{8(2 - \sqrt{5})}{(2 + \sqrt{5})(2 - \sqrt{5})}
\]

Simplifying the denominator:

\[
(2 + \sqrt{5})(2 - \sqrt{5}) = 2^2 - (\sqrt{5})^2 = 4 - 5 = -1
\]

Now, simplifying the expression:

\[
\frac{8(2 - \sqrt{5})}{-1} = -8(2 - \sqrt{5}) = -16 + 8\sqrt{5}
\]

So the simplified expression is:

\[
\boxed{-16 + 8\sqrt{5}}
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you find the conjugate of a binomial expression?
2. Why does multiplying by the conjugate help eliminate the square root in the denominator?
3. Can this method be applied to other expressions with radicals in the denominator?
4. How do you rationalize the denominator of an expression?
5. What are the steps to simplify an expression involving complex numbers?

**Tip:** When rationalizing a denominator with a square root, multiplying by the conjugate can help remove the radical.

Discover how to simplify the expression 8/(2+√5) by rationalizing the denominator using conjugates. This step-by-step solution covers important algebraic concepts, including radicals and the difference of squares theorem. Ideal for students in Grades 9-10.

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