Find an approximate value of the following corrected to 4 decimal places.
`root(3)(1002)-root(3)(998)`

Answer

To find the approximate value of \( \sqrt[3]{1002} - \sqrt[3]{998} \) corrected to four decimal places, we can use the binomial theorem for small values. Here’s how we can solve it step by step:

### Step 1: Rewrite the expression
We can express the cube roots in terms of a common base:
\[ \sqrt[3]{1002} - \sqrt[3]{998} = \sqrt[3]{1000 + 2} - \sqrt[3]{1000 - 2} \]

### Step 2: Factor out the common term
We can factor out \( \sqrt[3]{1000} \) (which is \( 10 \)):
\[ = \sqrt[3]{1000} \left( \sqrt[3]{1 + \frac{2}{1000}} - \sqrt[3]{1 - \frac{2}{1000}} \right) \]
\[ = 10 \left( \sqrt[3]{1 + 0.002} - \sqrt[3]{1 - 0.002} \right) \]

### Step 3: Use the binomial expansion
Using the binomial expansion for small \( x \):
\[ (1 + x)^n \approx 1 + nx + \frac{n(n-1)}{2}x^2 \]
For \( n = \frac{1}{3} \) and \( x = 0.002 \):
\[ \sqrt[3]{1 + 0.002} \approx 1 + \frac{1}{3}(0.002) + \frac{1/3 \cdot (1/3 - 1)}{2}(0.002)^2 \]
\[ \sqrt[3]{1 - 0.002} \approx 1 - \frac{1}{3}(0.002) + \frac{1/3 \cdot (1/3 - 1)}{2}(0.002)^2 \]

### Step 4: Calculate the expansions
Calculating the first few terms:
1. For \( \sqrt[3]{1 + 0.002} \):
\[ \approx 1 + \frac{1}{3}(0.002) + \frac{1/3 \cdot (-2/3)}{2}(0.002)^2 \]
\[ = 1 + 0.0006667 - 0.0000006667 \approx 1 + 0.0006667 \]

2. For \( \sqrt[3]{1 - 0.002} \):
\[ \approx 1 - \frac{1}{3}(0.002) + \frac{1/3 \cdot (-2/3)}{2}(0.002)^2 \]
\[ = 1 - 0.0006667 - 0.0000006667 \approx 1 - 0.0006667 \]

### Step 5: Subtract the two expansions
Now we can subtract the two expansions:
\[ \sqrt[3]{1 + 0.002} - \sqrt[3]{1 - 0.002} \approx (1 + 0.0006667) - (1 - 0.0006667) = 0.0006667 + 0.0006667 = 0.0013334 \]

### Step 6: Multiply by 10
Now, we multiply by 10:
\[ 10 \times 0.0013334 = 0.013334 \]

### Step 7: Round to four decimal places
Finally, rounding \( 0.013334 \) to four decimal places gives:
\[ \approx 0.0133 \]

### Final Answer
Thus, the approximate value of \( \sqrt[3]{1002} - \sqrt[3]{998} \) corrected to four decimal places is:
\[ \boxed{0.0133} \]