If p = 999, then the value of ` 3 sqrt( p ( p^(2) + 3 p + 3)+ 1)` is

Options

A. 1000
B. 999
C. 998
D. 1002

Answer

To solve the problem, we need to evaluate the expression \( 3 \sqrt{p(p^2 + 3p + 3) + 1} \) given that \( p = 999 \).

### Step-by-Step Solution:

1. **Substituting the value of \( p \)**:
\[ p = 999 \]
Substitute \( p \) into the expression:
\[ 3 \sqrt{999(999^2 + 3 \cdot 999 + 3) + 1} \]

2. **Calculating \( p^2 + 3p + 3 \)**:
First, calculate \( 999^2 \):
\[ 999^2 = 998001 \]
Now calculate \( 3 \cdot 999 \):
\[ 3 \cdot 999 = 2997 \]
Now add these values:
\[ 999^2 + 3 \cdot 999 + 3 = 998001 + 2997 + 3 = 1000001 \]

3. **Substituting back into the expression**:
Now substitute back into the expression:
\[ 3 \sqrt{999 \cdot 1000001 + 1} \]

4. **Calculating \( 999 \cdot 1000001 \)**:
\[ 999 \cdot 1000001 = 999000999 \]
Now add 1:
\[ 999000999 + 1 = 999001000 \]

5. **Taking the square root**:
Now we need to take the square root:
\[ \sqrt{999001000} \]
We can simplify this:
\[ \sqrt{999001000} = 999500 \]
(since \( 999001000 = (999500)^2 \))

6. **Final calculation**:
Now multiply by 3:
\[ 3 \cdot 999500 = 2998500 \]

Thus, the final value of the expression is:
\[ \boxed{2998500} \]