Evaluate the following : `(((12.5)^(3)+(7.5)^(3)))/((12.5)^(2) + (7.5)^(2) - 12.5 xx 7.5)`

Options

A. 20
B. 25
C. 30
D. 15

Answer

To evaluate the expression \(\frac{(12.5)^3 + (7.5)^3}{(12.5)^2 + (7.5)^2 - 12.5 \times 7.5}\), we can use the identity for the sum of cubes and simplify the expression step by step.

### Step 1: Identify the components
Let \(a = 12.5\) and \(b = 7.5\). We can rewrite the expression using these variables:
\[ \frac{a^3 + b^3}{a^2 + b^2 - ab} \]

### Step 2: Apply the sum of cubes identity
The identity for the sum of cubes states that:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
Using this identity, we can rewrite the numerator:
\[ a^3 + b^3 = (12.5 + 7.5)((12.5)^2 - 12.5 \times 7.5 + (7.5)^2) \]

### Step 3: Calculate \(a + b\)
Calculating \(a + b\):
\[ 12.5 + 7.5 = 20 \]

### Step 4: Calculate \(a^2 + b^2 - ab\)
Now, we need to calculate \(a^2 + b^2 - ab\):
1. Calculate \(a^2\):
\[ (12.5)^2 = 156.25 \]
2. Calculate \(b^2\):
\[ (7.5)^2 = 56.25 \]
3. Calculate \(ab\):
\[ 12.5 \times 7.5 = 93.75 \]
4. Now substitute these values into the expression:
\[ a^2 + b^2 - ab = 156.25 + 56.25 - 93.75 \]
\[ = 156.25 + 56.25 - 93.75 = 118.75 \]

### Step 5: Substitute back into the expression
Now substitute back into the original expression:
\[ \frac{(12.5)^3 + (7.5)^3}{(12.5)^2 + (7.5)^2 - 12.5 \times 7.5} = \frac{20 \times 118.75}{118.75} \]

### Step 6: Simplify the expression
The \(118.75\) in the numerator and denominator cancels out:
\[ = 20 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{20} \]