If `x + 1/x = 5`, then `x^6 + 1/(x^6)` is

Options

A. 12098
B. 12048
C. 14062
D. 12092

Answer

To solve the problem where \( x + \frac{1}{x} = 5 \) and we need to find \( x^6 + \frac{1}{x^6} \), we can follow these steps:

### Step 1: Square the given equation
We start with the equation:
\[ x + \frac{1}{x} = 5 \]
Now, we square both sides:
\[ \left(x + \frac{1}{x}\right)^2 = 5^2 \]
This expands to:
\[ x^2 + 2 + \frac{1}{x^2} = 25 \]
Subtracting 2 from both sides gives us:
\[ x^2 + \frac{1}{x^2} = 25 - 2 = 23 \]

### Step 2: Cube the result from Step 1
Now that we have \( x^2 + \frac{1}{x^2} = 23 \), we will cube this result:
\[ \left(x^2 + \frac{1}{x^2}\right)^3 = 23^3 \]
Expanding the left side using the formula \( (a + b)^3 = a^3 + b^3 + 3ab(a + b) \):
\[ x^6 + 3\left(x^2 \cdot \frac{1}{x^2}\right)(x^2 + \frac{1}{x^2}) + \frac{1}{x^6} = 23^3 \]
Since \( x^2 \cdot \frac{1}{x^2} = 1 \), we can simplify this to:
\[ x^6 + \frac{1}{x^6} + 3(x^2 + \frac{1}{x^2}) = 23^3 \]
Substituting \( x^2 + \frac{1}{x^2} = 23 \):
\[ x^6 + \frac{1}{x^6} + 3 \cdot 23 = 23^3 \]

### Step 3: Calculate \( 23^3 \)
Now we calculate \( 23^3 \):
\[ 23^3 = 23 \times 23 \times 23 = 529 \times 23 = 12167 \]

### Step 4: Solve for \( x^6 + \frac{1}{x^6} \)
Now we can substitute back into our equation:
\[ x^6 + \frac{1}{x^6} + 69 = 12167 \]
Subtracting 69 from both sides gives:
\[ x^6 + \frac{1}{x^6} = 12167 - 69 = 12098 \]

### Final Answer
Thus, the final result is:
\[ \boxed{12098} \]