`(63.5 xx 63.5 xx 63.5 + 36.5 xx 36.5 xx 36.5) /(6.35 xx 6.35 + 3.65 xx 3.65 - 6.35 xx 3.65`)` is equal to :

Options

A. 100
B. 1000
C. 1,00,000
D. 10000

Answer

To solve the expression \((63.5 \times 63.5 \times 63.5 + 36.5 \times 36.5 \times 36.5) /(6.35 \times 6.35 + 3.65 \times 3.65 - 6.35 \times 3.65)\), we can simplify it step by step.

### Step 1: Define Variables
Let:
- \( A = 63.5 \)
- \( B = 36.5 \)

### Step 2: Rewrite the Expression
The expression can be rewritten using \( A \) and \( B \):
\[ \frac{A^3 + B^3}{(A/10)^2 + (B/10)^2 - (A/10)(B/10)} \]

### Step 3: Simplify the Denominator
Calculating the denominator:
\[ \frac{A^2}{100} + \frac{B^2}{100} - \frac{AB}{100} = \frac{A^2 + B^2 - AB}{100} \]

### Step 4: Substitute Back into the Expression
Now substitute this back into the expression:
\[ \frac{A^3 + B^3}{\frac{A^2 + B^2 - AB}{100}} = \frac{100(A^3 + B^3)}{A^2 + B^2 - AB} \]

### Step 5: Use the Formula for \( A^3 + B^3 \)
Using the identity \( A^3 + B^3 = (A + B)(A^2 - AB + B^2) \):
\[ A^3 + B^3 = (A + B)(A^2 - AB + B^2) \]

### Step 6: Substitute the Identity
Now substituting this into the expression:
\[ \frac{100((A + B)(A^2 - AB + B^2))}{A^2 + B^2 - AB} \]

### Step 7: Cancel Out Terms
Notice that \( A^2 - AB + B^2 = (A^2 + B^2 - AB) \):
\[ \frac{100(A + B)(A^2 + B^2 - AB)}{A^2 + B^2 - AB} = 100(A + B) \]

### Step 8: Calculate \( A + B \)
Now calculate \( A + B \):
\[ A + B = 63.5 + 36.5 = 100 \]

### Step 9: Final Calculation
Thus, we have:
\[ 100(A + B) = 100 \times 100 = 10,000 \]

### Final Answer
The value of the expression is \( \boxed{10,000} \).