`(1.003)^(4)` is nearby equal to

Options

A. 1.012
B. 1.0012
C. 0.988
D. 1.003

Answer

To find the value of \( (1.003)^4 \) using the Binomial Theorem, we can follow these steps:

### Step 1: Rewrite the expression
We can express \( 1.003 \) as \( 1 + 0.003 \). Therefore, we have:
\[ (1.003)^4 = (1 + 0.003)^4 \]

### Step 2: Apply the Binomial Theorem
According to the Binomial Theorem, \( (1 + x)^n \) can be expanded as:
\[ (1 + x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \ldots \]
In our case, \( n = 4 \) and \( x = 0.003 \).

### Step 3: Calculate the first few terms
1. The first term is:
\[ 1 \]
2. The second term is:
\[ nx = 4 \cdot 0.003 = 0.012 \]
3. The third term is:
\[ \frac{n(n-1)}{2!}x^2 = \frac{4 \cdot 3}{2} \cdot (0.003)^2 = 6 \cdot 0.000009 = 0.000054 \]
4. The fourth term is:
\[ \frac{n(n-1)(n-2)}{3!}x^3 = \frac{4 \cdot 3 \cdot 2}{6} \cdot (0.003)^3 = 4 \cdot 0.000000027 = 0.000000108 \]

### Step 4: Sum the significant terms
Now we can sum the significant terms:
\[ (1.003)^4 \approx 1 + 0.012 + 0.000054 + 0.000000108 \]
Since the last two terms are very small, we can approximate:
\[ (1.003)^4 \approx 1 + 0.012 = 1.012 \]

### Final Result
Thus, the nearby value for \( (1.003)^4 \) is:
\[ \boxed{1.012} \]