If `sqrt(2)=1.414`, then find the value of `(1)/(sqrt(9)-sqrt(8))`.

Answer

To solve the problem \( \frac{1}{\sqrt{9} - \sqrt{8}} \) given that \( \sqrt{2} = 1.414 \), we will follow these steps:

### Step 1: Simplify the square roots
First, we simplify \( \sqrt{9} \) and \( \sqrt{8} \):
- \( \sqrt{9} = 3 \)
- \( \sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2} \)

So, we can rewrite the expression as:
\[ \frac{1}{3 - 2\sqrt{2}} \]

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

### Step 3: Expand the denominator
Using the difference of squares formula \( (a - b)(a + b) = a^2 - b^2 \):
- Here, \( a = 3 \) and \( b = 2\sqrt{2} \):
\[ (3 - 2\sqrt{2})(3 + 2\sqrt{2}) = 3^2 - (2\sqrt{2})^2 = 9 - 4 \cdot 2 = 9 - 8 = 1 \]

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

### Step 5: Substitute the value of \( \sqrt{2} \)
Now we substitute \( \sqrt{2} = 1.414 \):
\[ 3 + 2 \cdot 1.414 = 3 + 2.828 = 5.828 \]

### Final Answer
Thus, the value of \( \frac{1}{\sqrt{9} - \sqrt{8}} \) is:
\[ \boxed{5.828} \]