If `x+ (1)/(x) = sqrt3` find
`x^(48)+ (1)/(x^(48))`

Answer

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

1. **Given Equation**:
\[ x + \frac{1}{x} = \sqrt{3} \]

2. **Cube Both Sides**:
We will cube both sides of the equation:
\[ \left(x + \frac{1}{x}\right)^3 = \left(\sqrt{3}\right)^3 \]

3. **Expand Using the Formula**:
Using the identity \( (a + b)^3 = a^3 + b^3 + 3ab(a + b) \), we can rewrite the left side:
\[ x^3 + \frac{1}{x^3} + 3\left(x \cdot \frac{1}{x}\right)\left(x + \frac{1}{x}\right) = 3\sqrt{3} \]
Since \( x \cdot \frac{1}{x} = 1 \), this simplifies to:
\[ x^3 + \frac{1}{x^3} + 3\sqrt{3} = 3\sqrt{3} \]

4. **Isolate \( x^3 + \frac{1}{x^3} \)**:
Now, we can isolate \( x^3 + \frac{1}{x^3} \):
\[ x^3 + \frac{1}{x^3} = 3\sqrt{3} - 3\sqrt{3} = 0 \]

5. **Find \( x^6 + \frac{1}{x^6} \)**:
We can use the identity \( x^3 + \frac{1}{x^3} \) to find \( x^6 + \frac{1}{x^6} \):
\[ \left(x^3 + \frac{1}{x^3}\right)^2 = x^6 + \frac{1}{x^6} + 2 \]
Substituting \( x^3 + \frac{1}{x^3} = 0 \):
\[ 0^2 = x^6 + \frac{1}{x^6} + 2 \]
Therefore:
\[ x^6 + \frac{1}{x^6} = -2 \]

6. **Find \( x^{12} + \frac{1}{x^{12}} \)**:
Again, using the identity:
\[ \left(x^6 + \frac{1}{x^6}\right)^2 = x^{12} + \frac{1}{x^{12}} + 2 \]
Substituting \( x^6 + \frac{1}{x^6} = -2 \):
\[ (-2)^2 = x^{12} + \frac{1}{x^{12}} + 2 \]
Therefore:
\[ 4 = x^{12} + \frac{1}{x^{12}} + 2 \]
So:
\[ x^{12} + \frac{1}{x^{12}} = 2 \]

7. **Find \( x^{24} + \frac{1}{x^{24}} \)**:
Using the same identity:
\[ \left(x^{12} + \frac{1}{x^{12}}\right)^2 = x^{24} + \frac{1}{x^{24}} + 2 \]
Substituting \( x^{12} + \frac{1}{x^{12}} = 2 \):
\[ 2^2 = x^{24} + \frac{1}{x^{24}} + 2 \]
Therefore:
\[ 4 = x^{24} + \frac{1}{x^{24}} + 2 \]
So:
\[ x^{24} + \frac{1}{x^{24}} = 2 \]

8. **Find \( x^{48} + \frac{1}{x^{48}} \)**:
Finally, using the same identity:
\[ \left(x^{24} + \frac{1}{x^{24}}\right)^2 = x^{48} + \frac{1}{x^{48}} + 2 \]
Substituting \( x^{24} + \frac{1}{x^{24}} = 2 \):
\[ 2^2 = x^{48} + \frac{1}{x^{48}} + 2 \]
Therefore:
\[ 4 = x^{48} + \frac{1}{x^{48}} + 2 \]
So:
\[ x^{48} + \frac{1}{x^{48}} = 2 \]

Thus, the final answer is:
\[ \boxed{2} \]