Calculate the value of the expression: \(\sqrt{\frac{36}{\sqrt{5}-1}}-9\sqrt{5}\)

Let's break down this problem step-by-step.

1. Simplify the fraction inside the square root:
   

   First, we rationalize the denominator of the fraction \(\frac{36}{\sqrt{5}-1}\). To do this, we multiply the numerator and the denominator by the conjugate of the denominator, which is $\backslash (\backslash sqrt\{5\}\backslash )+1$.

   
   

   \[ \frac{36}{\sqrt{5}-1} \times \frac{\sqrt{5}+1}{\sqrt{5}+1} = \frac{36(\sqrt{5}+1)}{(\sqrt{5})^2 - 1^2} = \frac{36(\sqrt{5}+1)}{5-1} = \frac{36(\sqrt{5}+1)}{4} \]
   

   
   

   Now, we can simplify this further:

   
   

   \[ \frac{36(\sqrt{5}+1)}{4} = 9(\sqrt{5}+1) \]

2. Substitute back into the original expression:
   

   Now the expression inside the main square root becomes:

   
   

   \[ 9(\sqrt{5}+1) - 9\sqrt{5} \]

   
   

   Let's distribute the 9:

   
   

   \[ 9\sqrt{5} + 9 - 9\sqrt{5} \]

   
   

   The terms $$9\sqrt{5}$$ and $$-9\sqrt{5}$$ cancel each other out, leaving:

   
   

   \[ 9 \]

3. Calculate the final square root:
   

   So, the original expression simplifies to:

   
   

   \[ \sqrt{9} \]

   
   

   The square root of 9 is 3.

Ответ: 3