`1/(sqrt9-sqrt8)=?`

Options

A. `1/2(3-2sqrt2)`
B. `1/(3+2sqrt2)`
C. `(3-2sqrt2)`
D. `(3+2sqrt2)`

Answer

To solve the expression \( \frac{1}{\sqrt{9} - \sqrt{8}} \), we can follow these steps:

### Step 1: Simplify the square roots
First, we simplify the square roots in the expression:
\[ \sqrt{9} = 3 \quad \text{and} \quad \sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2} \]
So, we rewrite the expression:
\[ \frac{1}{\sqrt{9} - \sqrt{8}} = \frac{1}{3 - 2\sqrt{2}} \]

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

### Step 3: Calculate the denominator
Now we calculate the denominator using the difference of squares:
\[ (3 - 2\sqrt{2})(3 + 2\sqrt{2}) = 3^2 - (2\sqrt{2})^2 = 9 - 8 = 1 \]

### Step 4: Simplify the expression
Now, we can simplify the expression:
\[ \frac{3 + 2\sqrt{2}}{1} = 3 + 2\sqrt{2} \]

### Final Answer
Thus, the final answer is:
\[ \frac{1}{\sqrt{9} - \sqrt{8}} = 3 + 2\sqrt{2} \]

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