<Question>
Simplify the following expression to a simplified fraction: $$\sqrt{\dfrac{\dfrac{5}{\sqrt{80}}+\dfrac{\sqrt{845}}{9}+\sqrt{45}}{\sqrt5}}.$$
</Question>

<think>
<Plan_1>
Okay, so I need to simplify this expression: the square root of the whole fraction where the numerator is (5 over sqrt(80)) plus (sqrt(845) over 9) plus sqrt(45), and the denominator is sqrt(5). Hmm, that looks a bit complicated, but maybe if I break it down step by step, it will make more sense.

First, let me write down the expression again to make sure I have it right:
√[( (5/√80) + (√845/9) + √45 ) / √5]

Alright, so the main structure is a square root of a big fraction. The numerator of that big fraction has three terms: 5 over the square root of 80, plus square root of 845 over 9, plus square root of 45. The denominator of the big fraction is just sqrt(5). So, my plan is to simplify each of the terms in the numerator first, then combine them, and finally divide by sqrt(5) and take the square root at the end.

Let me start with simplifying each term in the numerator.

First term: 5 / sqrt(80). Let's see, sqrt(80) can be simplified.

<prompt_1.1> Simplify the first term in the numerator, 5 over sqrt(80). </prompt_1.1>

Second term: sqrt(845)/9. Let's simplify sqrt(845). Let me factor 845.

<prompt_1.2> Simplify the second term in the numerator, sqrt(845) over 9. </prompt_1.2>

Third term: sqrt(45). Let's simplify that.

<prompt_1.3> Simplify the third term in the numerator, sqrt(45). </prompt_1.3>
</Plan_1>

<Execution_1>
<execution_1.1>
sqrt(80) is sqrt(16*5) which is 4*sqrt(5). So, sqrt(80) = 4√5. Therefore, 5 / sqrt(80) becomes 5/(4√5). Hmm, maybe rationalize the denominator? If I multiply numerator and denominator by sqrt(5), that would give (5*sqrt(5))/(4*5) = sqrt(5)/4. Wait, let me check that again. 5 divided by 4√5 times sqrt(5)/sqrt(5) is (5*sqrt(5))/(4*5) because sqrt(5)*sqrt(5) is 5. So, the 5 in the numerator and the 5 in the denominator cancel out, leaving sqrt(5)/4. So, the first term simplifies to sqrt(5)/4.
</execution_1.1>

<execution_1.2>
845 divided by 5 is 169. So 845 is 5*169. And 169 is 13 squared. So sqrt(845) = sqrt(5*13²) = 13*sqrt(5). So sqrt(845)/9 is 13*sqrt(5)/9.
</execution_1.2>

<execution_1.3>
45 is 9*5, so sqrt(45) is 3*sqrt(5).
</execution_1.3>
</Execution_1>

<Plan_2>
Based on execution_1.1, execution_1.2, execution_1.3:
So now, substituting back into the numerator, we have:

<prompt_2.1> Combine the simplified terms in the numerator: sqrt(5)/4, 13*sqrt(5)/9, and 3*sqrt(5). </prompt_2.1>
</Plan_2>

<Execution_2>
<execution_2.1>
sqrt(5)/4 + 13*sqrt(5)/9 + 3*sqrt(5)

Let me factor out sqrt(5) from each term:

sqrt(5) * (1/4 + 13/9 + 3)

Now, let's compute the coefficients inside the parentheses. Let's handle the fractions first. The fractions are 1/4 and 13/9. To add them, I need a common denominator. The least common denominator of 4 and 9 is 36. So, converting each fraction:

1/4 = 9/36

13/9 = 52/36

So, 9/36 + 52/36 = 61/36

Then, we have +3. So, 61/36 + 3. Let's convert 3 to 36ths: 3 = 108/36. So, 61/36 + 108/36 = 169/36.

Therefore, the coefficients add up to 169/36. Therefore, the numerator simplifies to sqrt(5) * (169/36) = (169*sqrt(5))/36.
</execution_2.1>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Now, the entire expression inside the square root is this numerator divided by sqrt(5). So, we have:

<prompt_3.1> Simplify the entire expression inside the square root after dividing by sqrt(5). </prompt_3.1>
</Plan_3>

<Execution_3>
<execution_3.1>
[(169*sqrt(5))/36] / sqrt(5)

Dividing by sqrt(5) is the same as multiplying by 1/sqrt(5). So:

(169*sqrt(5))/36 * 1/sqrt(5) = 169/(36) * (sqrt(5)/sqrt(5)) = 169/36 * 1 = 169/36.

Because sqrt(5)/sqrt(5) is 1. So, the entire expression inside the square root simplifies to 169/36. Therefore, the original expression is sqrt(169/36).

Now, sqrt(169/36) is sqrt(169)/sqrt(36) = 13/6.
</execution_3.1>
</Execution_3>

<Plan_4>
Based on execution_3.1:
Wait, let me verify each step to make sure I didn't make a mistake.

<prompt_4.1> Verify the simplification of each term in the numerator. </prompt_4.1>
</Plan_4>

<Execution_4>
<execution_4.1>
Starting with the first term: 5/sqrt(80). sqrt(80) is 4*sqrt(5), so 5/(4*sqrt(5)). Multiply numerator and denominator by sqrt(5): (5*sqrt(5))/(4*5) = sqrt(5)/4. Correct.

Second term: sqrt(845)/9. 845 factors into 5*13², so sqrt(845) is 13*sqrt(5). So 13*sqrt(5)/9. Correct.

Third term: sqrt(45) is 3*sqrt(5). Correct.

Adding them up: sqrt(5)/4 + 13*sqrt(5)/9 + 3*sqrt(5). Factor out sqrt(5): coefficients are 1/4 + 13/9 + 3.

Convert to 36 denominator: 1/4 = 9/36, 13/9 = 52/36. So 9/36 + 52/36 = 61/36. Then 61/36 + 3. Since 3 is 108/36, total is 169/36. So coefficients total 169/36. Multiply by sqrt(5): (169*sqrt(5))/36. Correct.

Then divide by sqrt(5): (169*sqrt(5))/36 divided by sqrt(5) is (169*sqrt(5))/(36*sqrt(5)) = 169/36. Because sqrt(5)/sqrt(5) is 1. Then sqrt(169/36) is 13/6. Correct.
</execution_4.1>
</Execution_4>

<Plan_5>
Based on execution_4.1:
Let me check if 13/6 is in the simplest form.
13 is a prime number, and 6 is 2*3. 13 shares no common factors with 6, so yes, 13/6 is simplified.

So, the final answer should be 13/6.

But just to make sure, let me go through the steps again quickly.

<prompt_5.1> Quickly recheck the entire simplification process. </prompt_5.1>
</Plan_5>

<Execution_5>
<execution_5.1>
Original expression:
sqrt[ (5/sqrt(80) + sqrt(845)/9 + sqrt(45)) / sqrt(5) ]

Simplify each term:
5/sqrt(80) = sqrt(5)/4

sqrt(845)/9 = 13*sqrt(5)/9

sqrt(45) = 3*sqrt(5)

Sum them up: sqrt(5)/4 + 13*sqrt(5)/9 + 3*sqrt(5) = sqrt(5)*(1/4 + 13/9 + 3) = sqrt(5)*(169/36)

Divide by sqrt(5): (169/36)*sqrt(5)/sqrt(5) = 169/36

Take sqrt: sqrt(169/36) = 13/6.

Yes, all steps check out. So, the simplified fraction is 13/6.
</execution_5.1>
</Execution_5>

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