If `xx=sqrt(3)+sqrt(2),` then find `xx+1/x,x^(2)+(1)/(x^(2)),x^(3)+(1)/(x^(3)),x^(4)+(1)/(x^(4))`

Answer

To solve the problem step by step, we start with the given value of \( x \):

### Given:
\[ x = \sqrt{3} + \sqrt{2} \]

### Step 1: Find \( x + \frac{1}{x} \)

1. First, we need to find \( \frac{1}{x} \):
\[ \frac{1}{x} = \frac{1}{\sqrt{3} + \sqrt{2}} \]
To rationalize the denominator, we multiply the numerator and denominator by \( \sqrt{3} - \sqrt{2} \):
\[ \frac{1}{x} = \frac{\sqrt{3} - \sqrt{2}}{(\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{2})} = \frac{\sqrt{3} - \sqrt{2}}{3 - 2} = \sqrt{3} - \sqrt{2} \]

2. Now, we can find \( x + \frac{1}{x} \):
\[ x + \frac{1}{x} = (\sqrt{3} + \sqrt{2}) + (\sqrt{3} - \sqrt{2}) = 2\sqrt{3} \]

### Step 2: Find \( x^2 + \frac{1}{x^2} \)

1. We can use the identity:
\[ x^2 + \frac{1}{x^2} = \left( x + \frac{1}{x} \right)^2 - 2 \]
Substituting the value we found:
\[ x^2 + \frac{1}{x^2} = (2\sqrt{3})^2 - 2 = 12 - 2 = 10 \]

### Step 3: Find \( x^3 + \frac{1}{x^3} \)

1. We can use the identity:
\[ x^3 + \frac{1}{x^3} = \left( x + \frac{1}{x} \right)^3 - 3\left( x + \frac{1}{x} \right) \]
Substituting the value we found:
\[ x^3 + \frac{1}{x^3} = (2\sqrt{3})^3 - 3(2\sqrt{3}) = 24\sqrt{3} - 6\sqrt{3} = 18\sqrt{3} \]

### Step 4: Find \( x^4 + \frac{1}{x^4} \)

1. We can use the identity:
\[ x^4 + \frac{1}{x^4} = \left( x^2 + \frac{1}{x^2} \right)^2 - 2 \]
Substituting the value we found:
\[ x^4 + \frac{1}{x^4} = (10)^2 - 2 = 100 - 2 = 98 \]

### Final Results:
- \( x + \frac{1}{x} = 2\sqrt{3} \)
- \( x^2 + \frac{1}{x^2} = 10 \)
- \( x^3 + \frac{1}{x^3} = 18\sqrt{3} \)
- \( x^4 + \frac{1}{x^4} = 98 \)