If $x^{1/3}+x^{-1/3}=2,$ then $(x^{3}-x^{-3})$ is equal to:

Q: If $x^{1/3}+x^{-1/3}=2,$ then $(x^{3}-x^{-3})$ is equal to:

To solve the problem, we start with the given equation: \[ x^{1/3} + x^{-1/3} = 2. \] Let's set $ y = x^{1/3} $. Then, the equation becomes: \[ y + \frac{1}{y} = 2. \] To find $ y $, we multiply both sides by $ y $ to eliminate the fraction: \[ y^2 + 1 = 2y. \] Rearranging terms gives us: \[ y^2 - 2y + 1 = 0. \] This is a perfect square trinomial, which can be factored as: \[ (y - 1)^2 = 0. \] Solving for $ y $, we find: \[ y = 1. \] Since $ y = x^{1/3} $, we have: \[ x^{1/3} = 1. \] Cubing both sides, we find: \[ x = 1. \] Now, we need to find $ x^3 - x^{-3} $. Since $ x = 1 $, we have: \[ x^3 = 1^3 = 1 \] \[ x^{-3} = (1)^{-3} = 1. \] Therefore: \[ x^3 - x^{-3} = 1 - 1 = 0. \] Thus, the expression $ x^3 - x^{-3} $ evaluates to 0, which corresponds to Option 1. **Explanation for Other Options:** - **Option 2 (1):** This option is incorrect because, as shown, $ x^3 - x^{-3} = 0 $, not 1. - **Option 3 (2):** This option is incorrect because the calculation clearly shows that the result is 0, not 2. - **Option 4 (3):** This option is incorrect because the expression evaluates to 0, not 3. In conclusion, the correct answer is Option 1, as the calculations confirm that $ x^3 - x^{-3} = 0 $.

Question

Accepted Answer

To solve the problem, we start with the given equation:

$$ x^{1/3} + x^{-1/3} = 2. $$

Let's set $ y = x^{1/3} $. Then, the equation becomes:

$$ y + \frac{1}{y} = 2. $$

To find $ y $, we multiply both sides by $ y $ to eliminate the fraction:

$$ y^2 + 1 = 2y. $$

Rearranging terms gives us:

$$ y^2 - 2y + 1 = 0. $$

This is a perfect square trinomial, which can be factored as:

$$ (y - 1)^2 = 0. $$

Solving for $ y $, we find:

$$ y = 1. $$

Since $ y = x^{1/3} $, we have:

$$ x^{1/3} = 1. $$

Cubing both sides, we find:

$$ x = 1. $$

Now, we need to find $ x^3 - x^{-3} $. Since $ x = 1 $, we have:

$$ x^3 = 1^3 = 1 $$
$$ x^{-3} = (1)^{-3} = 1. $$

Therefore:

$$ x^3 - x^{-3} = 1 - 1 = 0. $$

Thus, the expression $ x^3 - x^{-3} $ evaluates to 0, which corresponds to Option 1.

**Explanation for Other Options:**

- **Option 2 (1):** This option is incorrect because, as shown, $ x^3 - x^{-3} = 0 $, not 1.
  
- **Option 3 (2):** This option is incorrect because the calculation clearly shows that the result is 0, not 2.
  
- **Option 4 (3):** This option is incorrect because the expression evaluates to 0, not 3.

In conclusion, the correct answer is Option 1, as the calculations confirm that $ x^3 - x^{-3} = 0 $.