If `x=1/(5+2sqrt6)`, the value of `1/x` is?

Options

A. `5+2sqrt6`
B. `5-2sqrt6`
C. `3+2sqrt3`
D. None of these

Answer

To find the value of \( \frac{1}{x} \) when \( x = \frac{1}{5 + 2\sqrt{6}} \), we can follow these steps:

### Step 1: Write the expression for \( \frac{1}{x} \)
Given \( x = \frac{1}{5 + 2\sqrt{6}} \), we can express \( \frac{1}{x} \) as:
\[ \frac{1}{x} = 5 + 2\sqrt{6} \]

### Step 2: Rationalize the denominator
To confirm this, we can multiply \( x \) by its conjugate to rationalize the denominator. The conjugate of \( 5 + 2\sqrt{6} \) is \( 5 - 2\sqrt{6} \).

### Step 3: Multiply numerator and denominator by the conjugate
We multiply both the numerator and denominator of \( x \) by \( 5 - 2\sqrt{6} \):
\[ x = \frac{1}{5 + 2\sqrt{6}} \cdot \frac{5 - 2\sqrt{6}}{5 - 2\sqrt{6}} = \frac{5 - 2\sqrt{6}}{(5 + 2\sqrt{6})(5 - 2\sqrt{6})} \]

### Step 4: Simplify the denominator
Now, we simplify the denominator using the difference of squares:
\[ (5 + 2\sqrt{6})(5 - 2\sqrt{6}) = 5^2 - (2\sqrt{6})^2 = 25 - 24 = 1 \]

### Step 5: Final expression for \( x \)
Thus, we have:
\[ x = 5 - 2\sqrt{6} \]

### Step 6: Calculate \( \frac{1}{x} \)
Now, substituting back to find \( \frac{1}{x} \):
\[ \frac{1}{x} = 5 + 2\sqrt{6} \]

### Conclusion
Therefore, the value of \( \frac{1}{x} \) is:
\[ \frac{1}{x} = 5 + 2\sqrt{6} \]

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