Answer :
To solve the given problem, we need to find the differences between the first term and the other terms. Let's express each term in their simplified forms and then calculate the differences step by step.
### Given Expressions
1. [tex]\( 11 \sqrt{45} \)[/tex]
2. [tex]\( 4 \sqrt{5} \)[/tex]
3. [tex]\( 7 \sqrt{40} \)[/tex]
4. [tex]\( 14 \sqrt{10} \)[/tex]
5. [tex]\( 29 \sqrt{5} \)[/tex]
6. [tex]\( 95 \sqrt{5} \)[/tex]
### Simplifying the Terms
First, we simplify each square root term:
#### Term 1: [tex]\( 11 \sqrt{45} \)[/tex]
[tex]\[ \sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3 \sqrt{5} \][/tex]
[tex]\[ 11 \sqrt{45} = 11 \times 3 \sqrt{5} = 33 \sqrt{5} \][/tex]
#### Term 2: [tex]\( 4 \sqrt{5} \)[/tex]
This term is already simplified.
#### Term 3: [tex]\( 7 \sqrt{40} \)[/tex]
[tex]\[ \sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2 \sqrt{10} \][/tex]
[tex]\[ 7 \sqrt{40} = 7 \times 2 \sqrt{10} = 14 \sqrt{10} \][/tex]
#### Term 4: [tex]\( 14 \sqrt{10} \)[/tex]
This term is already simplified.
#### Term 5: [tex]\( 29 \sqrt{5} \)[/tex]
This term is already simplified.
#### Term 6: [tex]\( 95 \sqrt{5} \)[/tex]
This term is already simplified.
### Computing the Differences
Now we find the differences between [tex]\( 33 \sqrt{5} \)[/tex] and the other terms:
#### Difference 1: [tex]\( 33 \sqrt{5} - 4 \sqrt{5} \)[/tex]
[tex]\[ 33 \sqrt{5} - 4 \sqrt{5} = (33 - 4) \sqrt{5} = 29 \sqrt{5} \][/tex]
#### Difference 2: [tex]\( 33 \sqrt{5} - 14 \sqrt{10} \)[/tex]
This term involves different square roots and remains:
[tex]\[ 33 \sqrt{5} - 14 \sqrt{10} \][/tex]
#### Difference 3: [tex]\( 33 \sqrt{5} - 14 \sqrt{10} \)[/tex]
This difference is the same as above:
[tex]\[ 33 \sqrt{5} - 14 \sqrt{10} \][/tex]
#### Difference 4: [tex]\( 33 \sqrt{5} - 29 \sqrt{5} \)[/tex]
[tex]\[ 33 \sqrt{5} - 29 \sqrt{5} = (33 - 29) \sqrt{5} = 4 \sqrt{5} \][/tex]
#### Difference 5: [tex]\( 33 \sqrt{5} - 95 \sqrt{5} \)[/tex]
[tex]\[ 33 \sqrt{5} - 95 \sqrt{5} = (33 - 95) \sqrt{5} = -62 \sqrt{5} \][/tex]
### Final Numerical Results
Evaluating these differences numerically:
1. [tex]\( 29 \sqrt{5} \approx 64.8459713474939 \)[/tex]
2. [tex]\( 33 \sqrt{5} - 14 \sqrt{10} \approx 29.518356015135744 \)[/tex]
3. [tex]\( 33 \sqrt{5} - 14 \sqrt{10} \approx 29.518356015135744 \)[/tex]
4. [tex]\( 4 \sqrt{5} \approx 8.94427190999916 \)[/tex]
5. [tex]\( -62 \sqrt{5} \approx -138.63621460498695 \)[/tex]
Hence, the results are:
1. [tex]\( 11 \sqrt{45} - 4 \sqrt{5} \approx 64.8459713474939 \)[/tex]
2. [tex]\( 11 \sqrt{45} - 7 \sqrt{40} \approx 29.518356015135744 \)[/tex]
3. [tex]\( 11 \sqrt{45} - 14 \sqrt{10} \approx 29.518356015135744 \)[/tex]
4. [tex]\( 11 \sqrt{45} - 29 \sqrt{5} \approx 8.94427190999916 \)[/tex]
5. [tex]\( 11 \sqrt{45} - 95 \sqrt{5} \approx -138.63621460498695 \)[/tex]
These are the numerical differences between the first term and each of the other terms.
### Given Expressions
1. [tex]\( 11 \sqrt{45} \)[/tex]
2. [tex]\( 4 \sqrt{5} \)[/tex]
3. [tex]\( 7 \sqrt{40} \)[/tex]
4. [tex]\( 14 \sqrt{10} \)[/tex]
5. [tex]\( 29 \sqrt{5} \)[/tex]
6. [tex]\( 95 \sqrt{5} \)[/tex]
### Simplifying the Terms
First, we simplify each square root term:
#### Term 1: [tex]\( 11 \sqrt{45} \)[/tex]
[tex]\[ \sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3 \sqrt{5} \][/tex]
[tex]\[ 11 \sqrt{45} = 11 \times 3 \sqrt{5} = 33 \sqrt{5} \][/tex]
#### Term 2: [tex]\( 4 \sqrt{5} \)[/tex]
This term is already simplified.
#### Term 3: [tex]\( 7 \sqrt{40} \)[/tex]
[tex]\[ \sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2 \sqrt{10} \][/tex]
[tex]\[ 7 \sqrt{40} = 7 \times 2 \sqrt{10} = 14 \sqrt{10} \][/tex]
#### Term 4: [tex]\( 14 \sqrt{10} \)[/tex]
This term is already simplified.
#### Term 5: [tex]\( 29 \sqrt{5} \)[/tex]
This term is already simplified.
#### Term 6: [tex]\( 95 \sqrt{5} \)[/tex]
This term is already simplified.
### Computing the Differences
Now we find the differences between [tex]\( 33 \sqrt{5} \)[/tex] and the other terms:
#### Difference 1: [tex]\( 33 \sqrt{5} - 4 \sqrt{5} \)[/tex]
[tex]\[ 33 \sqrt{5} - 4 \sqrt{5} = (33 - 4) \sqrt{5} = 29 \sqrt{5} \][/tex]
#### Difference 2: [tex]\( 33 \sqrt{5} - 14 \sqrt{10} \)[/tex]
This term involves different square roots and remains:
[tex]\[ 33 \sqrt{5} - 14 \sqrt{10} \][/tex]
#### Difference 3: [tex]\( 33 \sqrt{5} - 14 \sqrt{10} \)[/tex]
This difference is the same as above:
[tex]\[ 33 \sqrt{5} - 14 \sqrt{10} \][/tex]
#### Difference 4: [tex]\( 33 \sqrt{5} - 29 \sqrt{5} \)[/tex]
[tex]\[ 33 \sqrt{5} - 29 \sqrt{5} = (33 - 29) \sqrt{5} = 4 \sqrt{5} \][/tex]
#### Difference 5: [tex]\( 33 \sqrt{5} - 95 \sqrt{5} \)[/tex]
[tex]\[ 33 \sqrt{5} - 95 \sqrt{5} = (33 - 95) \sqrt{5} = -62 \sqrt{5} \][/tex]
### Final Numerical Results
Evaluating these differences numerically:
1. [tex]\( 29 \sqrt{5} \approx 64.8459713474939 \)[/tex]
2. [tex]\( 33 \sqrt{5} - 14 \sqrt{10} \approx 29.518356015135744 \)[/tex]
3. [tex]\( 33 \sqrt{5} - 14 \sqrt{10} \approx 29.518356015135744 \)[/tex]
4. [tex]\( 4 \sqrt{5} \approx 8.94427190999916 \)[/tex]
5. [tex]\( -62 \sqrt{5} \approx -138.63621460498695 \)[/tex]
Hence, the results are:
1. [tex]\( 11 \sqrt{45} - 4 \sqrt{5} \approx 64.8459713474939 \)[/tex]
2. [tex]\( 11 \sqrt{45} - 7 \sqrt{40} \approx 29.518356015135744 \)[/tex]
3. [tex]\( 11 \sqrt{45} - 14 \sqrt{10} \approx 29.518356015135744 \)[/tex]
4. [tex]\( 11 \sqrt{45} - 29 \sqrt{5} \approx 8.94427190999916 \)[/tex]
5. [tex]\( 11 \sqrt{45} - 95 \sqrt{5} \approx -138.63621460498695 \)[/tex]
These are the numerical differences between the first term and each of the other terms.
