To simplify the expression:

$\frac{\sqrt{10} + \sqrt{3}}{\sqrt{10} - \sqrt{3}}$

we'll multiply both the numerator and the denominator by the conjugate of the denominator, which is $\sqrt{10} + \sqrt{3}$. This is done to eliminate the square roots in the denominator.

Step 1: Multiply numerator and denominator by $\sqrt{10} + \sqrt{3}$

$\frac{\sqrt{10} + \sqrt{3}}{\sqrt{10} - \sqrt{3}} \times \frac{\sqrt{10} + \sqrt{3}}{\sqrt{10} + \sqrt{3}} = \frac{(\sqrt{10} + \sqrt{3})^2}{(\sqrt{10})^2 - (\sqrt{3})^2}$

Step 2: Apply the difference of squares in the denominator

The difference of squares formula is $a^2 - b^2 = (a - b)(a + b)$.

$= \frac{(\sqrt{10} + \sqrt{3})^2}{(\sqrt{10})^2 - (\sqrt{3})^2} = \frac{(\sqrt{10})^2 + 2\sqrt{10}\sqrt{3} + (\sqrt{3})^2}{10 - 3}$

Step 3: Simplify the numerator and denominator

$= \frac{10 + 2\sqrt{30} + 3}{7}$

$= \frac{13 + 2\sqrt{30}}{7}$

Thus, the simplified expression is:

$\frac{13 + 2\sqrt{30}}{7}$

Would you like more details on any step? Feel free to ask!

Here are 5 related questions:

1. How do you multiply binomials with square roots?
2. What is the significance of the conjugate in rationalizing denominators?
3. How can you simplify expressions involving radicals?
4. What are the steps to rationalize a complex denominator?
5. How do you apply the difference of squares formula?

Tip: Always check if the denominator can be simplified using the conjugate when dealing with square roots.