Find the value of the `(1)/(root(3)(999) )` correct to 4 decimal places

Answer

To find the value of \( \frac{1}{\sqrt[3]{999}} \) correct to four decimal places, we can follow these steps:

### Step 1: Rewrite the Expression
We start with the expression:
\[ \frac{1}{\sqrt[3]{999}} = \frac{1}{999^{1/3}} \]
We can express \( 999 \) as \( 1000 - 1 \):
\[ \frac{1}{\sqrt[3]{999}} = \frac{1}{(1000 - 1)^{1/3}} \]

### Step 2: Use the Binomial Expansion
Using the binomial expansion for \( (1 - x)^{n} \), where \( n = -\frac{1}{3} \) and \( x = \frac{1}{1000} \):
\[ (1 - x)^{-n} \approx 1 + nx + \frac{n(n+1)}{2!}x^2 + \ldots \]
Substituting \( x = \frac{1}{1000} \):
\[ (1 - \frac{1}{1000})^{-1/3} \approx 1 + \left(-\frac{1}{3}\right) \cdot \frac{1}{1000} + \frac{\left(-\frac{1}{3}\right)\left(-\frac{2}{3}\right)}{2} \cdot \left(\frac{1}{1000}\right)^2 \]

### Step 3: Calculate the First Two Terms
Calculating the first two terms:
1. The first term is \( 1 \).
2. The second term is:
\[ -\frac{1}{3} \cdot \frac{1}{1000} = -\frac{1}{3000} \]

### Step 4: Calculate the Second Order Term
Calculating the second order term:
\[ \frac{\left(-\frac{1}{3}\right)\left(-\frac{2}{3}\right)}{2} \cdot \left(\frac{1}{1000}\right)^2 = \frac{\frac{2}{9}}{2} \cdot \frac{1}{1000000} = \frac{1}{9 \cdot 1000000} = \frac{1}{9000000} \]

### Step 5: Combine the Terms
Now we combine the terms:
\[ (1 - \frac{1}{1000})^{-1/3} \approx 1 - \frac{1}{3000} + \frac{1}{9000000} \]
We can neglect the higher-order term \( \frac{1}{9000000} \) for our calculation up to four decimal places.

### Step 6: Approximate the Value
Thus, we have:
\[ \approx 1 - \frac{1}{3000} = 1 - 0.0003333 \approx 0.9996667 \]

### Step 7: Final Calculation
Now, we need to find:
\[ \frac{1}{10} \cdot (0.9996667) \approx 0.09996667 \]
Rounding this to four decimal places gives:
\[ \approx 0.1000 \]

### Final Answer
Thus, the value of \( \frac{1}{\sqrt[3]{999}} \) correct to four decimal places is:
\[ \boxed{0.1003} \]