Evaluate `(999)^(1//3)` upto 4 places of decimal.

Answer

To evaluate \( (999)^{1/3} \) up to four decimal places, we can use the binomial expansion method. Here’s a step-by-step solution:

### Step 1: Rewrite the expression
We can express \( 999 \) as \( 1000 - 1 \). Therefore, we can rewrite the expression as:
\[ (999)^{1/3} = (1000 - 1)^{1/3} \]

### Step 2: Apply the binomial expansion
Using the binomial expansion for \( (a - b)^n \), where \( a = 1000 \), \( b = 1 \), and \( n = \frac{1}{3} \), we have:
\[ (1000 - 1)^{1/3} = 1000^{1/3} \left(1 - \frac{1}{1000}\right)^{1/3} \]
Since \( 1000^{1/3} = 10 \), we can simplify this to:
\[ 10 \left(1 - \frac{1}{1000}\right)^{1/3} \]

### Step 3: Use the binomial approximation
For small \( x \), \( (1 - x)^n \approx 1 - nx \). Here, \( x = \frac{1}{1000} \) and \( n = \frac{1}{3} \):
\[ \left(1 - \frac{1}{1000}\right)^{1/3} \approx 1 - \frac{1}{3} \cdot \frac{1}{1000} \]
Calculating this gives:
\[ 1 - \frac{1}{3000} \]

### Step 4: Substitute back into the expression
Now we substitute back into our expression:
\[ (999)^{1/3} \approx 10 \left(1 - \frac{1}{3000}\right) = 10 - \frac{10}{3000} = 10 - \frac{1}{300} = 10 - 0.0033333 \]
Calculating this gives:
\[ 10 - 0.0033333 = 9.9966667 \]

### Step 5: Round to four decimal places
Rounding \( 9.9966667 \) to four decimal places gives:
\[ 9.9967 \]

### Final Answer
Thus, the value of \( (999)^{1/3} \) up to four decimal places is:
\[ \boxed{9.9967} \]