Answer :
Sure, let's go through each part of the question step by step to find the solutions.
### a) [tex]\(2 \sqrt{5} + 4 \sqrt{5} - \sqrt{5} + 3 \sqrt{5} =\)[/tex]
To combine like terms, we sum the coefficients of [tex]\(\sqrt{5}\)[/tex]:
[tex]\[ 2 \sqrt{5} + 4 \sqrt{5} - \sqrt{5} + 3 \sqrt{5} = (2 + 4 - 1 + 3) \cdot \sqrt{5} = 8 \cdot \sqrt{5} \][/tex]
Evaluating:
[tex]\[ 8 \cdot \sqrt{5} \approx 17.88854381999832 \][/tex]
### b) [tex]\(4 \sqrt{3} + 5 \sqrt{2} - 2 \sqrt{3} + 7 \sqrt{2} - \sqrt{2} =\)[/tex]
Again, we combine like terms:
[tex]\[ 4 \sqrt{3} - 2 \sqrt{3} = 2 \sqrt{3} \][/tex]
[tex]\[ 5 \sqrt{2} + 7 \sqrt{2} - \sqrt{2} = 11 \sqrt{2} \][/tex]
Summing these:
[tex]\[ 2 \sqrt{3} + 11 \sqrt{2} \][/tex]
Evaluating:
[tex]\[ 2 \sqrt{3} + 11 \sqrt{2} \approx 19.020450801241797 \][/tex]
### c) [tex]\(8 \sqrt[3]{4} + \sqrt[3]{4} - 6 \sqrt[8]{4} - 2 \sqrt[3]{4} =\)[/tex]
Combine terms involving [tex]\(\sqrt[3]{4}\)[/tex]:
[tex]\[ (8 + 1 - 2) \sqrt[3]{4} = 7 \sqrt[3]{4} \][/tex]
[tex]\[ - 6 \sqrt[8]{4} = -6 \sqrt[8]{4} \][/tex]
This gives:
[tex]\[ 7 \sqrt[3]{4} - 6 \sqrt[8]{4} \][/tex]
Evaluating:
[tex]\[ 7 (4^{1/3}) - 6 (4^{1/8}) \approx 3.97656467376107 \][/tex]
### d) [tex]\(9 \sqrt[3]{2} - 8 \sqrt{2} + 3 \sqrt{2} - 6 \sqrt[3]{2} + \sqrt{2} =\)[/tex]
Combine like terms:
[tex]\[ (9 - 6) \sqrt[3]{2} = 3 \sqrt[3]{2} \][/tex]
[tex]\[ (-8 + 3 + 1) \sqrt{2} = -4 \sqrt{2} \][/tex]
This gives:
[tex]\[ 3 \sqrt[3]{2} - 4 \sqrt{2} \][/tex]
Evaluating:
[tex]\[ 3 (2^{1/3}) - 4 (\sqrt{2}) \approx -1.877091099807761 \][/tex]
### e) [tex]\(3 \sqrt{27} + 5 \sqrt{48} + \sqrt{75} - 2 \sqrt{12} =\)[/tex]
Simplify the radicals:
[tex]\[ \sqrt{27} = 3 \sqrt{3} \][/tex]
[tex]\[ \sqrt{48} = 4 \sqrt{3} \][/tex]
[tex]\[ \sqrt{75} = 5 \sqrt{3} \][/tex]
[tex]\[ \sqrt{12} = 2 \sqrt{3} \][/tex]
This gives:
[tex]\[ 3\cdot 3 \sqrt{3} + 5\cdot 4 \sqrt{3} + 5 \sqrt{3} - 2\cdot 2 \sqrt{3} \][/tex]
[tex]\[ = 9 \sqrt{3} + 20 \sqrt{3} + 5 \sqrt{3} - 4 \sqrt{3} \][/tex]
[tex]\[ = 30 \sqrt{3} \][/tex]
Evaluating:
[tex]\[ 30 \sqrt{3} \approx 51.96152422706632 \][/tex]
### f) [tex]\(9 \sqrt[3]{16} - 3 \sqrt[3]{54} + 8 \sqrt[8]{250} - \sqrt[8]{2} =\)[/tex]
Simplify and evaluate each term:
[tex]\[ 9 \sqrt[3]{16} - 3 \sqrt[3]{54} + 8 \sqrt[8]{250} - \sqrt[8]{2} \][/tex]
Evaluating:
[tex]\[ 9(16^{1/3}) - 3(54^{1/3}) + 8(250^{1/8}) - (2^{1/8}) \approx 26.201418902931024 \][/tex]
### g) [tex]\(2 \sqrt[8]{24} + 5 \sqrt[8]{54} - 4 \sqrt[8]{81} + 6 \sqrt[8]{16} =\)[/tex]
Simplify and evaluate each term:
[tex]\[ 2 \sqrt[8]{24} + 5 \sqrt[8]{54} - 4 \sqrt[8]{81} + 6 \sqrt[8]{16} \][/tex]
Evaluating:
[tex]\[ 2(24^{1/8}) + 5(54^{1/8}) - 4(81^{1/8}) + 6(16^{1/8}) \approx 12.76481656364455 \][/tex]
### Summary of Results:
a) [tex]\( 2 \sqrt{5}+4 \sqrt{5}-\sqrt{5}+3 \sqrt{5} \approx 17.88854381999832 \)[/tex]
b) [tex]\( 4 \sqrt{3}+5 \sqrt{2}-2 \sqrt{3}+7 \sqrt{2}-\sqrt{2} \approx 19.020450801241797 \)[/tex]
c) [tex]\( 8 \sqrt[3]{4}+\sqrt[3]{4}-6 \sqrt[8]{4}-2 \sqrt[3]{4} \approx 3.97656467376107 \)[/tex]
d) [tex]\( 9 \sqrt[3]{2}-8 \sqrt{2}+3 \sqrt{2}-6 \sqrt[3]{2}+\sqrt{2} \approx -1.877091099807761 \)[/tex]
e) [tex]\( 3 \sqrt{27}+5 \sqrt{48}+\sqrt{75}-2 \sqrt{12} \approx 51.96152422706632 \)[/tex]
f) [tex]\( 9 \sqrt[3]{16}-3 \sqrt[3]{54}+8 \sqrt[8]{250}-\sqrt[8]{2} \approx 26.201418902931024 \)[/tex]
g) [tex]\( 2 \sqrt[8]{24}+5 \sqrt[8]{54}-4 \sqrt[8]{81}+6 \sqrt[8]{16} \approx 12.76481656364455 \)[/tex]
### a) [tex]\(2 \sqrt{5} + 4 \sqrt{5} - \sqrt{5} + 3 \sqrt{5} =\)[/tex]
To combine like terms, we sum the coefficients of [tex]\(\sqrt{5}\)[/tex]:
[tex]\[ 2 \sqrt{5} + 4 \sqrt{5} - \sqrt{5} + 3 \sqrt{5} = (2 + 4 - 1 + 3) \cdot \sqrt{5} = 8 \cdot \sqrt{5} \][/tex]
Evaluating:
[tex]\[ 8 \cdot \sqrt{5} \approx 17.88854381999832 \][/tex]
### b) [tex]\(4 \sqrt{3} + 5 \sqrt{2} - 2 \sqrt{3} + 7 \sqrt{2} - \sqrt{2} =\)[/tex]
Again, we combine like terms:
[tex]\[ 4 \sqrt{3} - 2 \sqrt{3} = 2 \sqrt{3} \][/tex]
[tex]\[ 5 \sqrt{2} + 7 \sqrt{2} - \sqrt{2} = 11 \sqrt{2} \][/tex]
Summing these:
[tex]\[ 2 \sqrt{3} + 11 \sqrt{2} \][/tex]
Evaluating:
[tex]\[ 2 \sqrt{3} + 11 \sqrt{2} \approx 19.020450801241797 \][/tex]
### c) [tex]\(8 \sqrt[3]{4} + \sqrt[3]{4} - 6 \sqrt[8]{4} - 2 \sqrt[3]{4} =\)[/tex]
Combine terms involving [tex]\(\sqrt[3]{4}\)[/tex]:
[tex]\[ (8 + 1 - 2) \sqrt[3]{4} = 7 \sqrt[3]{4} \][/tex]
[tex]\[ - 6 \sqrt[8]{4} = -6 \sqrt[8]{4} \][/tex]
This gives:
[tex]\[ 7 \sqrt[3]{4} - 6 \sqrt[8]{4} \][/tex]
Evaluating:
[tex]\[ 7 (4^{1/3}) - 6 (4^{1/8}) \approx 3.97656467376107 \][/tex]
### d) [tex]\(9 \sqrt[3]{2} - 8 \sqrt{2} + 3 \sqrt{2} - 6 \sqrt[3]{2} + \sqrt{2} =\)[/tex]
Combine like terms:
[tex]\[ (9 - 6) \sqrt[3]{2} = 3 \sqrt[3]{2} \][/tex]
[tex]\[ (-8 + 3 + 1) \sqrt{2} = -4 \sqrt{2} \][/tex]
This gives:
[tex]\[ 3 \sqrt[3]{2} - 4 \sqrt{2} \][/tex]
Evaluating:
[tex]\[ 3 (2^{1/3}) - 4 (\sqrt{2}) \approx -1.877091099807761 \][/tex]
### e) [tex]\(3 \sqrt{27} + 5 \sqrt{48} + \sqrt{75} - 2 \sqrt{12} =\)[/tex]
Simplify the radicals:
[tex]\[ \sqrt{27} = 3 \sqrt{3} \][/tex]
[tex]\[ \sqrt{48} = 4 \sqrt{3} \][/tex]
[tex]\[ \sqrt{75} = 5 \sqrt{3} \][/tex]
[tex]\[ \sqrt{12} = 2 \sqrt{3} \][/tex]
This gives:
[tex]\[ 3\cdot 3 \sqrt{3} + 5\cdot 4 \sqrt{3} + 5 \sqrt{3} - 2\cdot 2 \sqrt{3} \][/tex]
[tex]\[ = 9 \sqrt{3} + 20 \sqrt{3} + 5 \sqrt{3} - 4 \sqrt{3} \][/tex]
[tex]\[ = 30 \sqrt{3} \][/tex]
Evaluating:
[tex]\[ 30 \sqrt{3} \approx 51.96152422706632 \][/tex]
### f) [tex]\(9 \sqrt[3]{16} - 3 \sqrt[3]{54} + 8 \sqrt[8]{250} - \sqrt[8]{2} =\)[/tex]
Simplify and evaluate each term:
[tex]\[ 9 \sqrt[3]{16} - 3 \sqrt[3]{54} + 8 \sqrt[8]{250} - \sqrt[8]{2} \][/tex]
Evaluating:
[tex]\[ 9(16^{1/3}) - 3(54^{1/3}) + 8(250^{1/8}) - (2^{1/8}) \approx 26.201418902931024 \][/tex]
### g) [tex]\(2 \sqrt[8]{24} + 5 \sqrt[8]{54} - 4 \sqrt[8]{81} + 6 \sqrt[8]{16} =\)[/tex]
Simplify and evaluate each term:
[tex]\[ 2 \sqrt[8]{24} + 5 \sqrt[8]{54} - 4 \sqrt[8]{81} + 6 \sqrt[8]{16} \][/tex]
Evaluating:
[tex]\[ 2(24^{1/8}) + 5(54^{1/8}) - 4(81^{1/8}) + 6(16^{1/8}) \approx 12.76481656364455 \][/tex]
### Summary of Results:
a) [tex]\( 2 \sqrt{5}+4 \sqrt{5}-\sqrt{5}+3 \sqrt{5} \approx 17.88854381999832 \)[/tex]
b) [tex]\( 4 \sqrt{3}+5 \sqrt{2}-2 \sqrt{3}+7 \sqrt{2}-\sqrt{2} \approx 19.020450801241797 \)[/tex]
c) [tex]\( 8 \sqrt[3]{4}+\sqrt[3]{4}-6 \sqrt[8]{4}-2 \sqrt[3]{4} \approx 3.97656467376107 \)[/tex]
d) [tex]\( 9 \sqrt[3]{2}-8 \sqrt{2}+3 \sqrt{2}-6 \sqrt[3]{2}+\sqrt{2} \approx -1.877091099807761 \)[/tex]
e) [tex]\( 3 \sqrt{27}+5 \sqrt{48}+\sqrt{75}-2 \sqrt{12} \approx 51.96152422706632 \)[/tex]
f) [tex]\( 9 \sqrt[3]{16}-3 \sqrt[3]{54}+8 \sqrt[8]{250}-\sqrt[8]{2} \approx 26.201418902931024 \)[/tex]
g) [tex]\( 2 \sqrt[8]{24}+5 \sqrt[8]{54}-4 \sqrt[8]{81}+6 \sqrt[8]{16} \approx 12.76481656364455 \)[/tex]
