`a=9-4sqrt(5)` ,`sqrt(a)-1/sqrt(a)=?`

Explanation

To solve the problem where \( a = 9 - 4\sqrt{5} \) and we need to find \( \sqrt{a} - \frac{1}{\sqrt{a}} \), we can follow these steps:

### Step 1: Rewrite \( a \)
We start with the expression for \( a \):
\[ a = 9 - 4\sqrt{5} \]
We can express 9 as \( 5 + 4 \) to help us factor the expression:
\[ a = (5 + 4) - 4\sqrt{5} \]

### Step 2: Rearrange to form a perfect square
Next, we can rearrange \( a \) to form a perfect square:
\[ a = \sqrt{5}^2 + 4 - 4\sqrt{5} \]
This can be rewritten as:
\[ a = (\sqrt{5} - 2)^2 \]

### Step 3: Find \( \sqrt{a} \)
Now, we take the square root of \( a \):
\[ \sqrt{a} = \sqrt{(\sqrt{5} - 2)^2} = \sqrt{5} - 2 \]

### Step 4: Substitute into the expression
Now we substitute \( \sqrt{a} \) into the expression \( \sqrt{a} - \frac{1}{\sqrt{a}} \):
\[ \sqrt{a} - \frac{1}{\sqrt{a}} = (\sqrt{5} - 2) - \frac{1}{\sqrt{5} - 2} \]

### Step 5: Rationalize the denominator
To simplify \( \frac{1}{\sqrt{5} - 2} \), we multiply the numerator and denominator by the conjugate \( \sqrt{5} + 2 \):
\[ \frac{1}{\sqrt{5} - 2} \cdot \frac{\sqrt{5} + 2}{\sqrt{5} + 2} = \frac{\sqrt{5} + 2}{(\sqrt{5})^2 - 2^2} = \frac{\sqrt{5} + 2}{5 - 4} = \sqrt{5} + 2 \]

### Step 6: Combine the terms
Now we can combine the terms:
\[ \sqrt{a} - \frac{1}{\sqrt{a}} = (\sqrt{5} - 2) - (\sqrt{5} + 2) \]
This simplifies to:
\[ \sqrt{a} - \frac{1}{\sqrt{a}} = \sqrt{5} - 2 - \sqrt{5} - 2 = -4 \]

### Final Answer
Thus, the final result is:
\[ \sqrt{a} - \frac{1}{\sqrt{a}} = -4 \]