Solving Algebraic Simplification: The Sum of Cubes

This problem involves simplifying a complex-looking mathematical expression using a fundamental algebraic identity. We need to evaluate the expression:

$$ \frac{885 \times 885 \times 885 + 115 \times 115 \times 115}{885 \times 885 + 115 \times 115 - 885 \times 115} $$

Directly calculating the cubes and products would be very time-consuming. Instead, we can recognize that this expression fits a known algebraic pattern.

Leveraging the Sum of Cubes Algebraic Identity

The key to solving this problem efficiently is the algebraic identity for the sum of two cubes:

$$ a^3 + b^3 = (a+b)(a^2 - ab + b^2) $$

Let's compare this identity to our given expression. We can set:

• $a = 885$
• $b = 115$

Now, let's rewrite the expression using $a$ and $b$:

$$ \text{Expression} = \frac{a^3 + b^3}{a^2 + b^2 - ab} $$

Applying the Identity for Simplification

We can substitute the factored form of $a^3 + b^3$ into our expression:

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

Notice that the term $(a^2 - ab + b^2)$ appears in both the numerator and the denominator. Since $a=885$ and $b=115$, neither $a$ nor $b$ is zero, and the denominator $(a^2 + b^2 - ab)$ will not be zero. Therefore, we can cancel out this common term:

$$ \text{Expression} = a+b $$

Calculating the Final Result

Now, the problem simplifies to just adding the values of $a$ and $b$:

$$ \text{Result} = 885 + 115 $$

$$ \text{Result} = 1000 $$

Thus, the value of the given expression is 1000.