`((785 xx 785 xx 785+ 435 xx 435 xx 435)/((785)^2 + (435)^2-(785)(435)))` simplifies to :

Options

A. 350
B. 785
C. 1220
D. 1320

Answer

To simplify the expression \(\frac{785^3 + 435^3}{785^2 + 435^2 - 785 \cdot 435}\), we can follow these steps:

### Step 1: Identify Variables
Let \(a = 785\) and \(b = 435\). This allows us to rewrite the expression in terms of \(a\) and \(b\):
\[ \frac{a^3 + b^3}{a^2 + b^2 - ab} \]

### Step 2: Use the Formula for Sum of Cubes
Recall the algebraic identity for the sum of cubes:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
Thus, we can rewrite the numerator:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]

### Step 3: Simplify the Denominator
Now, we can also express the denominator:
\[ a^2 + b^2 - ab \]
This is already in a suitable form.

### Step 4: Substitute into the Expression
Substituting the expressions for the numerator and denominator into our original fraction gives:
\[ \frac{(a + b)(a^2 - ab + b^2)}{a^2 + b^2 - ab} \]

### Step 5: Cancel Common Terms
Notice that \(a^2 - ab + b^2\) can be rewritten as \(a^2 + b^2 - ab\) in the denominator. Therefore, we can cancel \(a^2 + b^2 - ab\) from both the numerator and denominator:
\[ = a + b \]

### Step 6: Substitute Back the Values of \(a\) and \(b\)
Now, substituting back the values of \(a\) and \(b\):
\[ = 785 + 435 \]

### Step 7: Perform the Addition
Calculating the sum:
\[ 785 + 435 = 1220 \]

### Final Answer
Thus, the simplified expression is:
\[ \frac{785^3 + 435^3}{785^2 + 435^2 - 785 \cdot 435} = 1220 \]