If `N=sqrt(9)+sqrt(6)`, then what is the value of 1/N?

Options

A. `sqrt(9)-sqrt(6)`
B. `3(sqrt(9) -sqrt(6))`
C. `(sqrt(9) - sqrt(6))//3`
D. `(sqrt(9) - sqrt(6))//4`

Answer

To find the value of \( \frac{1}{N} \) where \( N = \sqrt{9} + \sqrt{6} \), we can follow these steps:

### Step 1: Calculate \( N \)
First, we need to calculate the value of \( N \):
\[ N = \sqrt{9} + \sqrt{6} \]
Calculating the square roots:
\[ \sqrt{9} = 3 \quad \text{and} \quad \sqrt{6} \text{ remains as } \sqrt{6} \]
Thus,
\[ N = 3 + \sqrt{6} \]

### Step 2: Find \( \frac{1}{N} \)
Now, we need to find \( \frac{1}{N} \):
\[ \frac{1}{N} = \frac{1}{3 + \sqrt{6}} \]

### Step 3: Rationalize the Denominator
To simplify \( \frac{1}{3 + \sqrt{6}} \), we will rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is \( 3 - \sqrt{6} \):
\[ \frac{1}{3 + \sqrt{6}} \cdot \frac{3 - \sqrt{6}}{3 - \sqrt{6}} = \frac{3 - \sqrt{6}}{(3 + \sqrt{6})(3 - \sqrt{6})} \]

### Step 4: Simplify the Denominator
Now, we simplify the denominator using the difference of squares:
\[ (3 + \sqrt{6})(3 - \sqrt{6}) = 3^2 - (\sqrt{6})^2 = 9 - 6 = 3 \]

### Step 5: Final Expression
Now we can write:
\[ \frac{1}{N} = \frac{3 - \sqrt{6}}{3} \]
This can be separated into two fractions:
\[ \frac{1}{N} = 1 - \frac{\sqrt{6}}{3} \]

### Conclusion
Thus, the value of \( \frac{1}{N} \) is:
\[ \frac{1}{N} = 1 - \frac{\sqrt{6}}{3} \]