Okay, let's tackle this problem step by step. The expression we need to simplify is a bit complicated, so I should break it down into smaller parts. Let me start by understanding the structure of the expression.

The main expression is a square root of a fraction. The numerator of this fraction is another expression involving square roots and fractions, and the denominator is √5. So, the overall structure is √[(A + B + C)/√5], where A, B, and C are the individual terms in the numerator.

First, I need to simplify each of the terms in the numerator: 5/√80, √845/9, and √45. Then, add them together and divide by √5, and finally take the square root of the result. Let's start with each term individually.

Starting with the first term: 5/√80. I remember that when you have a square root in the denominator, it's often helpful to rationalize the denominator. To do that, multiply numerator and denominator by √80. Let's try that:

5/√80 = (5 * √80) / (√80 * √80) = (5√80)/80. 

Simplify √80. 80 is 16*5, so √80 = √(16*5) = 4√5. Therefore, substituting back:

(5 * 4√5)/80 = (20√5)/80 = √5/4. 

Okay, so the first term simplifies to √5/4. That's simpler. Let me note that as A = √5/4.

Next term: √845/9. Let's simplify √845. Let's factor 845. 845 divided by 5 is 169, and 169 is 13². So, 845 = 5 * 13². Therefore, √845 = √(5*13²) = 13√5. So, substituting back:

√845/9 = 13√5/9. Let's call this term B = 13√5/9.

Third term: √45. Let's simplify √45. 45 is 9*5, so √45 = √(9*5) = 3√5. Let's call this term C = 3√5.

So now, the numerator of the fraction inside the square root is A + B + C = √5/4 + 13√5/9 + 3√5. 

Since all terms have √5, we can factor that out. Let's factor √5:

√5*(1/4 + 13/9 + 3). 

Now, let's compute the coefficients inside the parentheses. Let's convert all fractions to have a common denominator. The denominators are 4, 9, and 1 (since 3 is 3/1). The least common multiple of 4 and 9 is 36. So, convert each fraction:

1/4 = 9/36,

13/9 = 52/36,

3 = 108/36.

Adding them together: 9/36 + 52/36 + 108/36 = (9 + 52 + 108)/36 = 169/36.

Therefore, the numerator becomes √5*(169/36) = (169√5)/36.

So now, the entire expression inside the square root is (169√5)/36 divided by √5. Let's write that as:

[(169√5)/36] / √5.

Dividing by √5 is the same as multiplying by 1/√5. So:

(169√5)/36 * 1/√5 = 169/(36*√5 * √5). Wait, no. Wait, hold on. Let's do this step carefully.

Actually, dividing by √5 is equivalent to multiplying numerator and denominator by √5 to rationalize. Wait, but in this case, the numerator is already (169√5)/36, and we are dividing that by √5. So:

[(169√5)/36] / √5 = 169√5 / (36 * √5) = 169 / 36.

Wait, because √5 in the numerator and denominator cancels out. So, that simplifies to 169/36. Then, the entire expression is the square root of 169/36.

√(169/36) = √169 / √36 = 13/6.

Wait, that seems straightforward. Let me verify each step again to make sure I didn't make a mistake.

Starting with the numerator:

First term: 5/√80 simplified to √5/4. Correct.

Second term: √845/9 simplified to 13√5/9. Correct.

Third term: √45 simplified to 3√5. Correct.

Adding them together: √5/4 + 13√5/9 + 3√5. Factor out √5: √5*(1/4 + 13/9 + 3). Correct.

Convert to common denominator 36:

1/4 = 9/36,

13/9 = 52/36,

3 = 108/36. Adding them: 9 + 52 + 108 = 169. So 169/36. Correct.

Multiply by √5: (169√5)/36. Then divide by √5: (169√5)/36 divided by √5 = 169/36. Then square root of that is 13/6. All steps check out.

Therefore, the simplified fraction is 13/6. Let me just make sure there are no calculation errors. Let's go through each term again quickly.

5/√80: √80 is 4√5, so 5/(4√5) = √5/4. Correct.

√845: 845 divided by 13 is 65, then 65 divided by 5 is 13. Wait, 13*65 = 845? Let me check: 13*60=780, 13*5=65, total 780+65=845. Yes, so 845=13*65. But 65 is 5*13, so 845=5*13*13=5*13². So √845=13√5. Correct.

√45=3√5. Correct.

So adding them: √5/4 +13√5/9 +3√5. Yes, factoring √5 gives √5*(1/4 +13/9 +3). Converting to 36 denominator: 9/36 +52/36 +108/36=169/36. Then √5*(169/36) divided by √5 is 169/36. Square root is 13/6. All steps are correct. 

I think that's the correct answer. Let me just confirm once more. If the original expression simplifies to 13/6, then squaring 13/6 gives (169/36), and multiplying by √5/√5 (since we divided by √5) gives back the original inside the square root. Wait, but in the steps above, when we divided (169√5)/36 by √5, we canceled out the √5, leaving 169/36. Then taking the square root gives 13/6. Yes, that's correct.

Alternatively, if I compute the entire expression numerically, let's check with approximate values.

Compute each term:

First term: 5/√80. √80≈8.944, so 5/8.944≈0.559.

Second term: √845≈29.068, divided by 9≈3.229.

Third term: √45≈6.708.

Adding them: 0.559 + 3.229 +6.708≈10.496.

Divide by √5≈2.236: 10.496/2.236≈4.700.

Then square root of 4.700≈2.167.

Now, 13/6≈2.166..., which matches the approximate value. So numerical approximation confirms the answer is 13/6.

Therefore, the simplified fraction is 13/6.
Therefore, the simplified fraction is $\boxed{\frac{13}{6}}$.
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