Simplify : $(a^{-1} + b^{-1}) ÷ (a^{-3}+b^{-3})$

$\frac{a^2b^2}{(a^2-ab+b^2)}$

We know that,

a³ + b³ = ( a + b ) ( a² + b² - ab )

So, $(a^{-1} + b^{-1}) ÷ (a^{-3}+b^{-3})$

= \(\frac{\frac{a + b}{ab}}{\frac{a^3 + b^3}{a^3b^3}}\)

= \(\frac{\frac{a + b}{ab}}{\frac{ ( a + b ) ( a^2+ b^2- ab )}{a^3b^3}}\)

= $\frac{a^2b^2}{(a^2-ab+b^2)}$