If `x^4 + 1//x^4 = 3842` then find the value of x + 1//x.

Options

A. 8
B. 6
C. 12
D. 7

Answer

To solve the problem, we need to find the value of \( x + \frac{1}{x} \) given that \( x^4 + \frac{1}{x^4} = 3842 \).

### Step 1: Relate \( x^4 + \frac{1}{x^4} \) to \( x^2 + \frac{1}{x^2} \)

We know that:
\[ x^4 + \frac{1}{x^4} = \left( x^2 + \frac{1}{x^2} \right)^2 - 2 \]
Let \( y = x^2 + \frac{1}{x^2} \). Then we can rewrite the equation as:
\[ x^4 + \frac{1}{x^4} = y^2 - 2 \]
So, we have:
\[ y^2 - 2 = 3842 \]

### Step 2: Solve for \( y^2 \)

Adding 2 to both sides:
\[ y^2 = 3842 + 2 = 3844 \]

### Step 3: Find \( y \)

Taking the square root of both sides:
\[ y = \sqrt{3844} \]
Calculating the square root:
\[ y = 62 \]

### Step 4: Relate \( y \) to \( x + \frac{1}{x} \)

Now, we need to find \( x + \frac{1}{x} \). We know that:
\[ x^2 + \frac{1}{x^2} = \left( x + \frac{1}{x} \right)^2 - 2 \]
Let \( z = x + \frac{1}{x} \). Then we can rewrite the equation as:
\[ x^2 + \frac{1}{x^2} = z^2 - 2 \]
Thus, we have:
\[ z^2 - 2 = 62 \]

### Step 5: Solve for \( z^2 \)

Adding 2 to both sides:
\[ z^2 = 62 + 2 = 64 \]

### Step 6: Find \( z \)

Taking the square root of both sides:
\[ z = \sqrt{64} \]
Calculating the square root:
\[ z = 8 \]

### Final Answer

Thus, the value of \( x + \frac{1}{x} \) is:
\[ \boxed{8} \]