If N = `sqrt7 - sqrt3`, then what is the value of (1/N)?

Options

A. `(sqrt7 + sqrt3)`
B. `(sqrt7 + sqrt3)//4`
C. `4(sqrt7 + sqrt3)`
D. `(sqrt7 + sqrt3)//2`

Answer

To find the value of \( \frac{1}{N} \) where \( N = \sqrt{7} - \sqrt{3} \), we can follow these steps:

### Step 1: Write the expression for \( \frac{1}{N} \)
We start with the expression:
\[ \frac{1}{N} = \frac{1}{\sqrt{7} - \sqrt{3}} \]

### Step 2: Rationalize the denominator
To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator, which is \( \sqrt{7} + \sqrt{3} \):
\[ \frac{1}{\sqrt{7} - \sqrt{3}} \cdot \frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} + \sqrt{3}} = \frac{\sqrt{7} + \sqrt{3}}{(\sqrt{7} - \sqrt{3})(\sqrt{7} + \sqrt{3})} \]

### Step 3: Simplify the denominator
Using the difference of squares formula \( (a - b)(a + b) = a^2 - b^2 \), we simplify the denominator:
\[ (\sqrt{7})^2 - (\sqrt{3})^2 = 7 - 3 = 4 \]

### Step 4: Write the final expression
Now we can write the expression as:
\[ \frac{\sqrt{7} + \sqrt{3}}{4} \]

### Step 5: Final answer
Thus, the value of \( \frac{1}{N} \) is:
\[ \frac{\sqrt{7} + \sqrt{3}}{4} \]

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