`(6.75 xx 6.75 xx 6.75 - 4.25 xx 4.25 xx 4.25)/(67.5 xx 67.5 + 42.5 xx 42.5 + 67.5 xx 42.5)` is equal to :

Options

A. A) 2.5
B. B) 0.25
C. C) 0.0025
D. D) 0.025

Answer

To solve the expression \((6.75 \times 6.75 \times 6.75 - 4.25 \times 4.25 \times 4.25) / (67.5 \times 67.5 + 42.5 \times 42.5 + 67.5 \times 42.5)\), we can use the formula for the difference of cubes and simplify the denominator.

### Step-by-step Solution:

1. **Identify the Cubes**:
We can rewrite the expression as:
\[ \frac{(6.75)^3 - (4.25)^3}{67.5^2 + 42.5^2 + 67.5 \times 42.5} \]

2. **Apply the Difference of Cubes Formula**:
The formula for the difference of cubes is:
\[ a^3 - b^3 = (a - b)(a^2 + b^2 + ab) \]
Here, \(a = 6.75\) and \(b = 4.25\). Thus, we can express the numerator as:
\[ (6.75 - 4.25)((6.75)^2 + (4.25)^2 + (6.75)(4.25)) \]

3. **Calculate \(a - b\)**:
\[ 6.75 - 4.25 = 2.5 \]

4. **Calculate \(a^2 + b^2 + ab\)**:
- Calculate \( (6.75)^2 \):
\[ (6.75)^2 = 45.5625 \]
- Calculate \( (4.25)^2 \):
\[ (4.25)^2 = 18.0625 \]
- Calculate \( (6.75)(4.25) \):
\[ (6.75)(4.25) = 28.6875 \]
- Now, sum these values:
\[ 45.5625 + 18.0625 + 28.6875 = 92.3125 \]

5. **Substitute Back into the Expression**:
Now, the numerator becomes:
\[ 2.5 \times 92.3125 \]

6. **Calculate the Numerator**:
\[ 2.5 \times 92.3125 = 230.78125 \]

7. **Calculate the Denominator**:
Now, calculate the denominator:
\[ 67.5^2 + 42.5^2 + 67.5 \times 42.5 \]
- Calculate \( (67.5)^2 \):
\[ (67.5)^2 = 4556.25 \]
- Calculate \( (42.5)^2 \):
\[ (42.5)^2 = 1806.25 \]
- Calculate \( (67.5)(42.5) \):
\[ (67.5)(42.5) = 2868.75 \]
- Now, sum these values:
\[ 4556.25 + 1806.25 + 2868.75 = 9231.25 \]

8. **Final Calculation**:
Now, substitute back into the expression:
\[ \frac{230.78125}{9231.25} \]

9. **Perform the Division**:
\[ \frac{230.78125}{9231.25} = 0.025 \]

### Final Answer:
Thus, the value of the expression is:
\[ \boxed{0.025} \]