To simplify the expression

$\frac{\sqrt{3} \cdot \sqrt{87}}{\sqrt{3} - 8},$

we can proceed as follows:

Step 1: Simplify the numerator

The numerator is $\sqrt{3} \cdot \sqrt{87}$. We can simplify this using the property of square roots:

$\sqrt{3} \cdot \sqrt{87} = \sqrt{3 \times 87}.$

Calculating $3 \times 87$:

$3 \times 87 = 261,$

so the numerator simplifies to:

$\sqrt{261}.$

Step 2: Rewrite the simplified expression

The expression now becomes:

$\frac{\sqrt{261}}{\sqrt{3} - 8}.$

Step 3: Rationalize the denominator

To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator, which is $\sqrt{3} + 8$:

$\frac{\sqrt{261}}{\sqrt{3} - 8} \times \frac{\sqrt{3} + 8}{\sqrt{3} + 8} = \frac{\sqrt{261} \cdot (\sqrt{3} + 8)}{(\sqrt{3} - 8)(\sqrt{3} + 8)}.$

Step 4: Simplify the denominator

Using the difference of squares, $(a - b)(a + b) = a^2 - b^2$:

$(\sqrt{3} - 8)(\sqrt{3} + 8) = (\sqrt{3})^2 - 8^2 = 3 - 64 = -61.$

Step 5: Simplify the entire expression

Now the expression is:

$\frac{\sqrt{261} \cdot (\sqrt{3} + 8)}{-61}.$

So, the simplified form of the given expression is:

$\frac{\sqrt{261} (\sqrt{3} + 8)}{-61}.$

Would you like further details, or do you have any questions?

Here are some related questions for further exploration:

1. How do you simplify square roots of products like $\sqrt{a} \cdot \sqrt{b}$?
2. What is the purpose of rationalizing a denominator?
3. How do you simplify complex fractions involving square roots?
4. How do you multiply expressions with square roots?
5. What are conjugates, and how are they used in simplifying fractions?

Tip: Rationalizing the denominator helps eliminate square roots or other irrational numbers from the bottom of the fraction, making expressions easier to work with.