The value of $\frac{6.35 × 6.35 × 6.35 + 3.65 × 3.65 × 3.65}{63.5 × 63.5+ 36.5 × 36.5 -63.5 × 36.5}$ is equal to:

0.1

a³ + b³ = ( a + b ) ( a² + b² - ab )

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

Given, 

$\frac{6.35 × 6.35 × 6.35 + 3.65 × 3.65 × 3.65}{63.5 × 63.5+ 36.5 × 36.5 -63.5 × 36.5}$

On comparing this equation with a³ + b³ = ( a + b ) ( a² + b² - ab ) we can conclude that 

a = 6.35 

b = 3.65

So the value of a + b = 6.35  + 3.65 = 10

a + b  = \(\frac{10}{100}\) = 0.1 (Divided by 100 because of decimal)