Evaluate : `15^(3)+10^(3)-25^(3)`

Options

A. `-11350`
B. `-11550`
C. `-11250`
D. `-11200`

Explanation

To evaluate the expression \( 15^3 + 10^3 - 25^3 \), we can utilize the identity for the sum of cubes and the properties of polynomials. Let's break it down step by step.

### Step 1: Identify the expression
We start with the expression:
\[ 15^3 + 10^3 - 25^3 \]

### Step 2: Recognize the identity
We can use the identity for the sum of cubes:
\[ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc) \]
In our case, we can set:
- \( a = 15 \)
- \( b = 10 \)
- \( c = -25 \)

### Step 3: Calculate \( a + b + c \)
Now, we calculate:
\[ a + b + c = 15 + 10 - 25 = 0 \]

### Step 4: Apply the identity
Since \( a + b + c = 0 \), we can simplify our expression using the identity:
\[ a^3 + b^3 + c^3 = 3abc \]
Thus, we need to calculate \( 3abc \):
\[ abc = 15 \cdot 10 \cdot (-25) \]

### Step 5: Calculate \( abc \)
Now, calculate \( abc \):
\[ abc = 15 \cdot 10 = 150 \]
\[ abc = 150 \cdot (-25) = -3750 \]

### Step 6: Calculate \( 3abc \)
Now, we calculate \( 3abc \):
\[ 3abc = 3 \cdot (-3750) = -11250 \]

### Step 7: Conclusion
Thus, the value of \( 15^3 + 10^3 - 25^3 \) is:
\[ \boxed{-11250} \]