Simplify: `((361)^(3)+(139)^(3))/((361)^(2)-(361xx139)+(139)^(2))`

Answer

To simplify the expression \(\frac{(361)^3 + (139)^3}{(361)^2 - (361 \times 139) + (139)^2}\), we can use the identity for the sum of cubes and the difference of squares.

### Step 1: Identify \(a\) and \(b\)
Let:
- \(a = 361\)
- \(b = 139\)

### Step 2: Apply the sum of cubes identity
The identity for the sum of cubes states that:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
Applying this identity, we have:
\[ (361)^3 + (139)^3 = (361 + 139)((361)^2 - (361)(139) + (139)^2) \]

### Step 3: Substitute into the expression
Now substituting back into the original expression:
\[ \frac{(361)^3 + (139)^3}{(361)^2 - (361 \times 139) + (139)^2} = \frac{(361 + 139)((361)^2 - (361)(139) + (139)^2)}{(361)^2 - (361 \times 139) + (139)^2} \]

### Step 4: Cancel the common terms
The numerator and denominator have a common factor of \((361)^2 - (361)(139) + (139)^2\), so we can cancel these terms:
\[ = 361 + 139 \]

### Step 5: Calculate the final result
Now, we just need to add \(361\) and \(139\):
\[ 361 + 139 = 500 \]

Thus, the simplified expression is:
\[ \boxed{500} \]