The value of `(1001)^3` is

Options

A. `1003003001`
B. `100303001`
C. `100300301`
D. 103003001

Answer

To find the value of \( (1001)^3 \), we can use the algebraic identity for the cube of a sum, which is given by:

\[ (a + b)^3 = a^3 + b^3 + 3ab(a + b) \]

### Step-by-Step Solution:

1. **Identify \( a \) and \( b \)**:
Let \( a = 1000 \) and \( b = 1 \). Therefore, we can express \( 1001 \) as \( a + b \).

2. **Apply the formula**:
Using the identity, we have:
\[ (1001)^3 = (1000 + 1)^3 = 1000^3 + 1^3 + 3 \cdot 1000 \cdot 1 \cdot (1000 + 1) \]

3. **Calculate \( 1000^3 \)**:
\[ 1000^3 = 1000 \times 1000 \times 1000 = 10^3 \times 10^3 \times 10^3 = 10^9 = 1000000000 \]

4. **Calculate \( 1^3 \)**:
\[ 1^3 = 1 \]

5. **Calculate \( 3 \cdot 1000 \cdot 1 \cdot (1000 + 1) \)**:
First, calculate \( (1000 + 1) \):
\[ 1000 + 1 = 1001 \]
Now calculate \( 3 \cdot 1000 \cdot 1 \cdot 1001 \):
\[ 3 \cdot 1000 \cdot 1 \cdot 1001 = 3000 \cdot 1001 \]

6. **Calculate \( 3000 \cdot 1001 \)**:
\[ 3000 \cdot 1001 = 3000 \cdot (1000 + 1) = 3000000 + 3000 = 3003000 \]

7. **Combine all the parts**:
Now, we add all the calculated parts together:
\[ (1001)^3 = 1000000000 + 1 + 3003000 \]
\[ = 1000000000 + 3003001 = 1003003001 \]

### Final Answer:
Thus, the value of \( (1001)^3 \) is \( 1003003001 \).