If `x = (7+ 4 sqrt(3))` then `(x+(1)/(x))= ?`

Options

A. `8 sqrt(2)`
B. `14`
C. `49`
D. `48`

Answer

To solve the problem where \( x = 7 + 4\sqrt{3} \) and we need to find \( x + \frac{1}{x} \), we can follow these steps:

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

Given \( x = 7 + 4\sqrt{3} \), we need to calculate \( \frac{1}{x} \):
\[ \frac{1}{x} = \frac{1}{7 + 4\sqrt{3}} \]

### Step 2: Rationalize the denominator

To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator, which is \( 7 - 4\sqrt{3} \):
\[ \frac{1}{7 + 4\sqrt{3}} \cdot \frac{7 - 4\sqrt{3}}{7 - 4\sqrt{3}} = \frac{7 - 4\sqrt{3}}{(7 + 4\sqrt{3})(7 - 4\sqrt{3})} \]

### Step 3: Simplify the denominator using the difference of squares

Using the difference of squares formula \( (a + b)(a - b) = a^2 - b^2 \):
\[ (7 + 4\sqrt{3})(7 - 4\sqrt{3}) = 7^2 - (4\sqrt{3})^2 = 49 - 48 = 1 \]
Thus, we have:
\[ \frac{1}{x} = 7 - 4\sqrt{3} \]

### Step 4: Add \( x \) and \( \frac{1}{x} \)

Now we can find \( x + \frac{1}{x} \):
\[ x + \frac{1}{x} = (7 + 4\sqrt{3}) + (7 - 4\sqrt{3}) \]

### Step 5: Combine like terms

When we combine the terms:
\[ = 7 + 4\sqrt{3} + 7 - 4\sqrt{3} = 7 + 7 + (4\sqrt{3} - 4\sqrt{3}) = 14 + 0 = 14 \]

### Final Answer

Thus, the value of \( x + \frac{1}{x} \) is:
\[ \boxed{14} \]