We are asked to find the value of the expression $\left(\frac{x^2 + y^2 - xy}{x^3 + y^3}\right)$ given $x = 12$ and $y = 7$.

Evaluating the Expression with Given Values

Substitute the values $x=12$ and $y=7$ into the expression.

Numerator: $x^2 + y^2 - xy$

$12^2 + 7^2 - (12)(7)$

$144 + 49 - 84$

$193 - 84 = 109$

Using Algebraic Factorization for Denominator

The denominator is $x^3 + y^3$. We can use the sum of cubes factorization formula:

$x^3 + y^3 = (x+y)(x^2 - xy + y^2)$

Notice that the term $(x^2 - xy + y^2)$ is exactly the numerator we calculated.

Simplifying the Expression

Now substitute the calculated numerator and the factored denominator back into the original expression:

$\left(\frac{x^2 + y^2 - xy}{x^3 + y^3}\right) = \left(\frac{109}{(x+y)(x^2 - xy + y^2)}\right)$

Substitute the value of the numerator $(109)$ and the value of $(x+y)$:

$x+y = 12 + 7 = 19$

So the expression becomes:

$\frac{109}{(19)(109)}$

Cancel out the common factor $109$ from the numerator and the denominator:

$\frac{1}{19}$

Final Value Calculation

The value of the expression is $\frac{1}{19}$.