If `x=(4+sqrt(15))^(1//3)+(4-sqrt(15))^(1//3)` , then show that `x^(3)-3x-8=0` .

Answer

To solve the problem, we need to show that if \( x = (4 + \sqrt{15})^{1/3} + (4 - \sqrt{15})^{1/3} \), then \( x^3 - 3x - 8 = 0 \).

### Step-by-Step Solution:

1. **Let \( a = (4 + \sqrt{15})^{1/3} \) and \( b = (4 - \sqrt{15})^{1/3} \)**:
\[ x = a + b \]

**Hint**: Define \( a \) and \( b \) for simplicity in calculations.

2. **Cube both sides**:
\[ x^3 = (a + b)^3 \]

**Hint**: Remember the identity \( (A + B)^3 = A^3 + B^3 + 3AB(A + B) \).

3. **Apply the identity**:
\[ x^3 = a^3 + b^3 + 3ab(a + b) \]

**Hint**: Substitute \( x \) back into the equation after applying the identity.

4. **Calculate \( a^3 \) and \( b^3 \)**:
\[ a^3 = 4 + \sqrt{15}, \quad b^3 = 4 - \sqrt{15} \]
\[ a^3 + b^3 = (4 + \sqrt{15}) + (4 - \sqrt{15}) = 8 \]

**Hint**: The cube of a cube root returns the original number.

5. **Calculate \( ab \)**:
\[ ab = ((4 + \sqrt{15})(4 - \sqrt{15}))^{1/3} = (4^2 - (\sqrt{15})^2)^{1/3} = (16 - 15)^{1/3} = 1^{1/3} = 1 \]

**Hint**: Use the difference of squares to simplify the multiplication.

6. **Substitute back into the equation**:
\[ x^3 = 8 + 3(1)(x) = 8 + 3x \]

**Hint**: Substitute \( ab \) and \( a + b \) into the equation.

7. **Rearrange the equation**:
\[ x^3 - 3x - 8 = 0 \]

**Hint**: Move all terms to one side to form a standard polynomial equation.

### Final Result:
We have shown that \( x^3 - 3x - 8 = 0 \), which is what we needed to prove.