Answer :

Certainly! Let's break down the problem step by step:

We start with the expression:
[tex]\[ \sqrt{25+24} + \left[ \left(5^3 - 100\right) + \left(\sqrt{25} + \sqrt{822}\right) + 5 \right] \][/tex]

### Step 1: Simplify the Square Root Expression
First, simplify [tex]\(\sqrt{25 + 24}\)[/tex]:
[tex]\[ 25 + 24 = 49 \][/tex]
[tex]\[ \sqrt{49} = 7 \][/tex]

So, [tex]\(\sqrt{25+24} = 7\)[/tex].

### Step 2: Simplify Inside the Brackets
Next, simplify the expression inside the brackets, starting with:
[tex]\[ 5^3 - 100 \][/tex]

Calculate [tex]\(5^3\)[/tex]:
[tex]\[ 5^3 = 125 \][/tex]

Then, subtract 100:
[tex]\[ 125 - 100 = 25 \][/tex]

We continue by simplifying [tex]\(\sqrt{25}\)[/tex] and [tex]\(\sqrt{822}\)[/tex]:
[tex]\[ \sqrt{25} = 5 \][/tex]

We represent [tex]\(\sqrt{822}\)[/tex] as an approximate value:
[tex]\[ \sqrt{822} \approx 28.67 \][/tex]

So the expression inside the brackets becomes:
[tex]\[ (25) + (5) + (28.67) + 5 \][/tex]

### Step 3: Sum Inside the Brackets
Let's add everything inside the brackets:
[tex]\[ 25 + 5 + 28.67 + 5 = 63.67 \][/tex]

### Step 4: Combine Results
Finally, add the results from outside and inside the brackets:
[tex]\[ 7 + 63.67 = 70.67 \][/tex]

### Conclusion
The final result of the expression is:
[tex]\[ 70.67 \][/tex]