We are tasked with simplifying the expression:

$\frac{5 + \sqrt{10}}{8 + \sqrt{10}}$

We want to express this as a single fraction in simplest radical form with a rational denominator. To do this, we rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator.

Step 1: Multiply by the conjugate of the denominator

The conjugate of $8 + \sqrt{10}$ is $8 - \sqrt{10}$. So, we multiply both the numerator and the denominator by $8 - \sqrt{10}$:

$\frac{5 + \sqrt{10}}{8 + \sqrt{10}} \times \frac{8 - \sqrt{10}}{8 - \sqrt{10}} = \frac{(5 + \sqrt{10})(8 - \sqrt{10})}{(8 + \sqrt{10})(8 - \sqrt{10})}$

Step 2: Simplify the denominator

The denominator is a difference of squares:

$(8 + \sqrt{10})(8 - \sqrt{10}) = 8^2 - (\sqrt{10})^2 = 64 - 10 = 54$

Step 3: Expand the numerator

Now, expand the product in the numerator $(5 + \sqrt{10})(8 - \sqrt{10})$:

$(5 + \sqrt{10})(8 - \sqrt{10}) = 5(8) + 5(-\sqrt{10}) + \sqrt{10}(8) + \sqrt{10}(-\sqrt{10})$

Simplifying each term:

$5(8) = 40$ $5(-\sqrt{10}) = -5\sqrt{10}$ $\sqrt{10}(8) = 8\sqrt{10}$ $\sqrt{10}(-\sqrt{10}) = -(\sqrt{10})^2 = -10$

Now, sum these terms:

$40 - 5\sqrt{10} + 8\sqrt{10} - 10 = 30 + 3\sqrt{10}$

Step 4: Write the final expression

Thus, the numerator becomes $30 + 3\sqrt{10}$ and the denominator is $54$. So, the expression simplifies to:

$\frac{30 + 3\sqrt{10}}{54}$

Step 5: Simplify the fraction

We can factor out a 3 from the numerator:

$\frac{30 + 3\sqrt{10}}{54} = \frac{3(10 + \sqrt{10})}{54}$

Now, divide both the numerator and the denominator by 3:

$\frac{10 + \sqrt{10}}{18}$

Final Answer:

The expression simplifies to:

$\frac{10 + \sqrt{10}}{18}$

Would you like further clarification or additional details on any step?

Here are 5 related questions to deepen understanding:

1. How do you find the conjugate of a binomial involving a square root?
2. What is the significance of rationalizing the denominator?
3. How do you simplify expressions involving square roots?
4. How does the difference of squares formula apply to binomials?
5. Can you generalize the method used here to other fractions involving square roots?

Tip: Always remember to factor and simplify after rationalizing the denominator—it helps make the final expression as simple as possible.