If $x=\sqrt[3]{28}$ and $y=\sqrt[3]{27}$ then the value of $x+y-\frac{1}{x^{2}+xy+y^{2}}$ is:

Question

Accepted Answer

To solve the problem and justify why Option 2 is correct, we need to evaluate the expression $ x + y - \frac{1}{x^2 + xy + y^2} $ given that $ x = \sqrt[3]{28} $ and $ y = \sqrt[3]{27} $.

1. **Calculate $ x + y $:**

- $ x = \sqrt[3]{28} $ and $ y = \sqrt[3]{27} $.
   - Since $ \sqrt[3]{27} = 3 $, we have $ y = 3 $.
   - Therefore, $ x + y = \sqrt[3]{28} + 3 $.

2. **Evaluate $ x^2 + xy + y^2 $:**

- $ x^2 = (\sqrt[3]{28})^2 = \sqrt[3]{28^2} = \sqrt[3]{784} $.
   - $ xy = \sqrt[3]{28} \times 3 = 3\sqrt[3]{28} $.
   - $ y^2 = 3^2 = 9 $.
   - Thus, $ x^2 + xy + y^2 = \sqrt[3]{784} + 3\sqrt[3]{28} + 9 $.

3. **Simplify the expression:**

- The expression we need to evaluate is $ x + y - \frac{1}{x^2 + xy + y^2} $.
   - Substituting the values, we have:
     $$
     x + y - \frac{1}{x^2 + xy + y^2} = \sqrt[3]{28} + 3 - \frac{1}{\sqrt[3]{784} + 3\sqrt[3]{28} + 9}
     $$

4. **Approximate the values:**

- $ \sqrt[3]{28} \approx 3.036 $ (since $ 3^3 = 27 $ and $ 4^3 = 64 $, $ \sqrt[3]{28} $ is slightly more than 3).
   - Therefore, $ x + y \approx 3.036 + 3 = 6.036 $.

5. **Evaluate the denominator:**

- The denominator $ x^2 + xy + y^2 $ is a positive number, and its exact value is not needed for the approximation since it is clear that it is greater than 1.

6. **Conclusion:**

- Since $ x + y \approx 6.036 $ and the term $\frac{1}{x^2 + xy + y^2}$ is a small positive number, the expression $ x + y - \frac{1}{x^2 + xy + y^2} $ is slightly less than 6.036.
   - Therefore, the closest integer value is 6.

Thus, the correct answer is Option 2, which is 6.

**Why other options are incorrect:**

- **Option 1 (2):** This value is too low given that $ x + y $ itself is approximately 6.036.
- **Option 3 (5):** This value is also too low because the expression $ x + y - \frac{1}{x^2 + xy + y^2} $ is slightly less than 6.036, not as low as 5.
- **Option 4 (1):** This is far too low, as $ x + y $ alone is much greater than 1.

In conclusion, Option 2 is the correct answer because it accurately reflects the approximate value of the expression given the values of $ x $ and $ y $.