Alright, let's break this problem down step by step. We need to add the following terms:
[tex]\[ 8 \sqrt{20} + 5 \][/tex]
[tex]\[ 17 \sqrt{5} \][/tex]
[tex]\[ 8 \sqrt{25} \][/tex]
[tex]\[ 9 \sqrt{25} \][/tex]
[tex]\[ 15 \sqrt{5} \][/tex]
### Step 1: Simplify each term.
1. Term 1: [tex]\( 8 \sqrt{20} + 5 \)[/tex]
- Simplify [tex]\(\sqrt{20}\)[/tex]:
[tex]\[
20 = 4 \times 5 \quad \text{so} \quad \sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2 \sqrt{5}
\][/tex]
- Thus, [tex]\( 8 \sqrt{20} = 8 \times 2 \sqrt{5} = 16 \sqrt{5} \)[/tex].
- So, [tex]\( 8 \sqrt{20} + 5 = 16 \sqrt{5} + 5 \)[/tex].
2. Term 2: [tex]\( 17 \sqrt{5} \)[/tex]
- This term is already in its simplest form: [tex]\( 17 \sqrt{5} \)[/tex].
3. Term 3: [tex]\( 8 \sqrt{25} \)[/tex]
- Simplify [tex]\(\sqrt{25}\)[/tex]:
[tex]\[
\sqrt{25} = 5
\][/tex]
- Thus, [tex]\( 8 \sqrt{25} = 8 \times 5 = 40 \)[/tex].
4. Term 4: [tex]\( 9 \sqrt{25} \)[/tex]
- Simplify [tex]\(\sqrt{25}\)[/tex]:
[tex]\[
\sqrt{25} = 5
\][/tex]
- Thus, [tex]\( 9 \sqrt{25} = 9 \times 5 = 45 \)[/tex].
5. Term 5: [tex]\( 15 \sqrt{5} \)[/tex]
- This term is already in its simplest form: [tex]\( 15 \sqrt{5} \)[/tex].
### Step 2: Combine the like terms.
#### Terms involving [tex]\( \sqrt{5} \)[/tex]:
[tex]\[ 16 \sqrt{5} + 17 \sqrt{5} + 15 \sqrt{5} \][/tex]
Combining these, we get:
[tex]\[ (16 + 17 + 15) \sqrt{5} = 48 \sqrt{5} \][/tex]
#### Constant terms:
[tex]\[ 5 + 40 + 45 \][/tex]
Combining these, we get:
[tex]\[ 5 + 40 + 45 = 90 \][/tex]
### Step 3: Add all the simplified terms to get the final result.
[tex]\[
48 \sqrt{5} + 90
\][/tex]
The individual values of the terms, for reference are:
[tex]\[ 16 \sqrt{5} + 5 \approx 40.777 \][/tex]
[tex]\[ 17 \sqrt{5} \approx 38.013 \][/tex]
[tex]\[ 8 \sqrt{25} = 40 \][/tex]
[tex]\[ 9 \sqrt{25} = 45 \][/tex]
[tex]\[ 15 \sqrt{5} \approx 33.541 \][/tex]
The sum involving [tex]\( \sqrt{5} \)[/tex] and constant terms are:
[tex]\[ 48 \sqrt{5} \approx 107.331 \][/tex]
[tex]\[ Constant terms sum = 90 \][/tex]
In conclusion, putting everything together, the final result is:
[tex]\[
48 \sqrt{5} + 90 \approx 197.331
\][/tex]
So, the detailed step-by-step addition of all the terms results in:
[tex]\[
48 \sqrt{5} + 90 \approx 197.331
\][/tex]
