Calculate `(1001)^(1//3)`.

Options

A. 10.00333
B. 10
C. 10.0333
D. 100

Explanation

To calculate \( (1001)^{1/3} \), we can use the binomial approximation method. Here’s a step-by-step solution:

### Step 1: Rewrite the expression
We start by rewriting \( 1001 \) as \( 1000 + 1 \):
\[ (1001)^{1/3} = (1000 + 1)^{1/3} \]

### Step 2: Factor out the cube root of 1000
We know that \( 1000 = 10^3 \), so we can express the cube root of \( 1000 \):
\[ (1000 + 1)^{1/3} = (10^3 + 1)^{1/3} \]

### Step 3: Use the binomial approximation
Using the binomial approximation \( (a + b)^n \approx a^n + n \cdot a^{n-1} \cdot b \) when \( b \) is small compared to \( a \), we set \( a = 1000 \) and \( b = 1 \) with \( n = \frac{1}{3} \):
\[ (1000 + 1)^{1/3} \approx 1000^{1/3} + \frac{1}{3} \cdot 1000^{-2/3} \cdot 1 \]

### Step 4: Calculate \( 1000^{1/3} \)
Calculating \( 1000^{1/3} \):
\[ 1000^{1/3} = 10 \]

### Step 5: Calculate \( 1000^{-2/3} \)
Next, we calculate \( 1000^{-2/3} \):
\[ 1000^{-2/3} = \frac{1}{(1000^{2/3})} = \frac{1}{(10^2)} = \frac{1}{100} = 0.01 \]

### Step 6: Substitute back into the approximation
Now we substitute back into our approximation:
\[ (1000 + 1)^{1/3} \approx 10 + \frac{1}{3} \cdot 0.01 \]

### Step 7: Calculate the additional term
Calculating \( \frac{1}{3} \cdot 0.01 \):
\[ \frac{1}{3} \cdot 0.01 = \frac{0.01}{3} \approx 0.003333 \]

### Step 8: Combine the results
Adding this to \( 10 \):
\[ (1001)^{1/3} \approx 10 + 0.003333 = 10.003333 \]

### Final Result
Thus, the value of \( (1001)^{1/3} \) is approximately:
\[ \boxed{10.003333} \]