The number obtained on rationalising the denominator of `(1)/(sqrt(7) - 2)` is

Options

A. `(sqrt(7) + 2)/(3)`
B. `(sqrt(2) - 2)/(3)`
C. `(sqrt(7) + 2)/(5)`
D. `(sqrt(7) + 2)/(45)`

Explanation

To rationalize the denominator of the expression \( \frac{1}{\sqrt{7} - 2} \), we follow these steps:

### Step 1: Identify the expression
We start with the expression:
\[ \frac{1}{\sqrt{7} - 2} \]

### Step 2: Multiply by the conjugate
To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator, which is \( \sqrt{7} + 2 \):
\[ \frac{1}{\sqrt{7} - 2} \cdot \frac{\sqrt{7} + 2}{\sqrt{7} + 2} \]

### Step 3: Simplify the expression
Now, we simplify the expression:
\[ = \frac{\sqrt{7} + 2}{(\sqrt{7} - 2)(\sqrt{7} + 2)} \]

### Step 4: Calculate the denominator
Next, we calculate the denominator using the difference of squares:
\[ (\sqrt{7} - 2)(\sqrt{7} + 2) = (\sqrt{7})^2 - (2)^2 = 7 - 4 = 3 \]

### Step 5: Write the final expression
Now we can write the simplified expression:
\[ = \frac{\sqrt{7} + 2}{3} \]

Thus, the number obtained by rationalizing the denominator of \( \frac{1}{\sqrt{7} - 2} \) is:
\[ \frac{\sqrt{7} + 2}{3} \]

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