Find the mean proportion of $\frac{a^3+b^3}{a-b}$ and $\frac{a^2-b^2}{a^2-a b+b^2}$.

a + b

Using direct formula,

Mean Proportion = \(\sqrt {a\; \times\; b }\)

                            = \(\sqrt { (\frac{a^3+b^3}{a-b})\; \times\; (\frac{a^2-b^2}{a^2-a b+b^2})}\) 

                            = \(\sqrt {\frac{ (a + b)(a²  - ab + b² )}{(a - b)}\; \times\; \frac{(a+b)(a-b)}{a^2-a b+b^2} }\)   

(using the formula a³ + b³ = (a + b)(a²  - ab + b² ) and a² - b² = (a + b)(a - b))    

                            = ( a + b)