This expression can be simplified algebraically. The given problem is:

$\frac{a\sqrt{a} - b\sqrt{b}}{\sqrt{a} - \sqrt{b}} + \sqrt{ab}.$

Step-by-Step Simplification:

1. Simplify the fraction: $\frac{a\sqrt{a} - b\sqrt{b}}{\sqrt{a} - \sqrt{b}}$

   The numerator $a\sqrt{a} - b\sqrt{b}$ can be expressed as: $\sqrt{a}(a) - \sqrt{b}(b),$ which is already factored partially.

   The denominator $\sqrt{a} - \sqrt{b}$ suggests a rationalization approach.

2. Rationalize the denominator (multiply numerator and denominator by $\sqrt{a} + \sqrt{b}$): $\frac{a\sqrt{a} - b\sqrt{b}}{\sqrt{a} - \sqrt{b}} \cdot \frac{\sqrt{a} + \sqrt{b}}{\sqrt{a} + \sqrt{b}}.$

   The denominator becomes: $(\sqrt{a})^2 - (\sqrt{b})^2 = a - b.$

   The numerator becomes: $(a\sqrt{a} - b\sqrt{b})(\sqrt{a} + \sqrt{b}) = a\sqrt{a}\sqrt{a} + a\sqrt{a}\sqrt{b} - b\sqrt{b}\sqrt{a} - b\sqrt{b}\sqrt{b}.$

   Simplify each term: $a^2 + a\sqrt{ab} - b\sqrt{ab} - b^2.$

   Combine: $a^2 - b^2 + (a - b)\sqrt{ab}.$

   So the fraction becomes: $\frac{a^2 - b^2 + (a - b)\sqrt{ab}}{a - b}.$

3. Simplify further: Split the terms: $\frac{a^2 - b^2}{a - b} + \frac{(a - b)\sqrt{ab}}{a - b}.$

   The first term simplifies using the difference of squares: $\frac{a^2 - b^2}{a - b} = a + b.$

   The second term simplifies: $\frac{(a - b)\sqrt{ab}}{a - b} = \sqrt{ab}.$

   So the simplified fraction is: $(a + b) + \sqrt{ab}.$

4. Add the remaining term ($+ \sqrt{ab}$): Combine the terms: $(a + b) + \sqrt{ab} + \sqrt{ab}.$

   This simplifies to: $a + b + 2\sqrt{ab}.$

Final Answer:

$a + b + 2\sqrt{ab}.$

Let me know if you need additional clarification! 😊

Related Questions:

1. How does the process of rationalizing a denominator work in general?
2. Why is the difference of squares helpful in simplifying this problem?
3. What happens if $a = b$ in the original expression?
4. Can you generalize this result for similar types of expressions?
5. How would this change if additional terms were added to the numerator?

Tip:

Always look for common algebraic patterns like factoring or the difference of squares to simplify complex fractions effectively.