Sure, let's go through each of the expressions one by one in detail:
1. [tex]\[ 18 \sqrt{6} + 9 \sqrt{6} \][/tex]
When you add two terms that have the same square root component, you can simply add the coefficients.
[tex]\[ 18 \sqrt{6} + 9 \sqrt{6} = (18 + 9) \sqrt{6} = 27 \sqrt{6} \][/tex]
Numerically:
[tex]\[ 27 \sqrt{6} \approx 66.1362230551458 \][/tex]
2. [tex]\[ 23 \sqrt{11} - 2 \sqrt{11} \][/tex]
When you subtract terms that have the same square root component, you can simply subtract the coefficients.
[tex]\[ 23 \sqrt{11} - 2 \sqrt{11} = (23 - 2) \sqrt{11} = 21 \sqrt{11} \][/tex]
Numerically:
[tex]\[ 21 \sqrt{11} \approx 69.6491205974634 \][/tex]
3. [tex]\[ 35 \sqrt{2} + \sqrt{32} \][/tex]
First, simplify [tex]\(\sqrt{32}\)[/tex]:
[tex]\[ \sqrt{32} = \sqrt{16 \cdot 2} = \sqrt{16} \cdot \sqrt{2} = 4 \sqrt{2} \][/tex]
Then, add the terms with the same square root component:
[tex]\[ 35 \sqrt{2} + 4 \sqrt{2} = (35 + 4) \sqrt{2} = 39 \sqrt{2} \][/tex]
Numerically:
[tex]\[ 39 \sqrt{2} \approx 55.15432893255071 \][/tex]
4. [tex]\[ 48 \sqrt{5} - 2 \sqrt{20} \][/tex]
First, simplify [tex]\(\sqrt{20}\)[/tex]:
[tex]\[ \sqrt{20} = \sqrt{4 \cdot 5} = \sqrt{4} \cdot \sqrt{5} = 2 \sqrt{5} \][/tex]
Then, substitute and simplify:
[tex]\[ 48 \sqrt{5} - 2 \sqrt{20} = 48 \sqrt{5} - 2 \cdot 2 \sqrt{5} = 48 \sqrt{5} - 4 \sqrt{5} = (48 - 4) \sqrt{5} = 44 \sqrt{5} \][/tex]
Numerically:
[tex]\[ 44 \sqrt{5} \approx 98.38699100999075 \][/tex]
5. [tex]\[ 5 - 2 \sqrt{13} + \sqrt{13} \][/tex]
Combine like terms with the same square root component:
[tex]\[ 5 - 2 \sqrt{13} + \sqrt{13} = 5 - (2 - 1) \sqrt{13} = 5 - \sqrt{13} \][/tex]
Numerically:
[tex]\[ 5 - \sqrt{13} \approx 1.3944487245360109 \][/tex]
These are the simplified and numerical forms of each expression.
