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 expansion method for approximating cube roots. Here’s a step-by-step solution:

### Step 1: Rewrite the expression
We can express 1001 as:
\[ 1001 = 1000 + 1 \]
Thus, we can write:
\[ (1001)^{1/3} = (1000 + 1)^{1/3} \]

### Step 2: Apply the binomial expansion
Using the binomial expansion for \( (a + b)^n \), where \( a = 1000 \), \( b = 1 \), and \( n = \frac{1}{3} \):
\[ (a + b)^n \approx a^n + \frac{n \cdot a^{n-1} \cdot b}{1!} \]
We can ignore higher-order terms since \( b \) is small compared to \( a \).

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

### Step 4: Calculate the first-order term
Now calculate the first-order term:
\[ \frac{1/3 \cdot 1000^{(1/3) - 1} \cdot 1}{1!} = \frac{1/3 \cdot 1000^{-2/3} \cdot 1}{1} = \frac{1}{3 \cdot 10^2} = \frac{1}{300} \]

### Step 5: Combine the results
Now combine the results:
\[ (1001)^{1/3} \approx 10 + \frac{1}{300} \]
To express this in decimal form:
\[ \frac{1}{300} \approx 0.00333 \]
Thus:
\[ (1001)^{1/3} \approx 10 + 0.00333 = 10.00333 \]

### Final Result
Therefore, the value of \( (1001)^{1/3} \) is approximately:
\[ \boxed{10.003} \]

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