Answer :
Para encontrar la raíz cuadrada descomponiendo los números en sus factores primos, seguiremos el enfoque de la descomposición factorial.
a. [tex]\(\sqrt{1225}\)[/tex]
1. [tex]\(1225 \div 5 = 245\)[/tex]
2. [tex]\(245 \div 5 = 49\)[/tex]
3. [tex]\(49 = 7 \times 7\)[/tex]
[tex]\[ 1225 = 5 \times 5 \times 7 \times 7 = (5 \times 7)^2 \][/tex]
Por lo tanto:
[tex]\[ \sqrt{1225} = 35 \][/tex]
b. [tex]\(\sqrt{1681}\)[/tex]
1. [tex]\(1681 = 41 \times 41\)[/tex]
[tex]\[ 1681 = (41)^2 \][/tex]
Por lo tanto:
[tex]\[ \sqrt{1681} = 41 \][/tex]
c. [tex]\(\sqrt{400}\)[/tex]
1. [tex]\(400 \div 2 = 200\)[/tex]
2. [tex]\(200 \div 2 = 100\)[/tex]
3. [tex]\(100 \div 2 = 50\)[/tex]
4. [tex]\(50 \div 2 = 25\)[/tex]
5. [tex]\(25 = 5 \times 5\)[/tex]
[tex]\[ 400 = 2 \times 2 \times 2 \times 2 \times 5 \times 5 = (2 \times 2 \times 5)^2 = (20)^2 \][/tex]
Por lo tanto:
[tex]\[ \sqrt{400} = 20 \][/tex]
d. [tex]\(\sqrt{6561}\)[/tex]
1. [tex]\(6561 \div 3 = 2187\)[/tex]
2. [tex]\(2187 \div 3 = 729\)[/tex]
3. [tex]\(729 \div 3 = 243\)[/tex]
4. [tex]\(243 \div 3 = 81\)[/tex]
5. [tex]\(81 \div 3 = 27\)[/tex]
6. [tex]\(27 \div 3 = 9\)[/tex]
7. [tex]\(9 \div 3 = 3\)[/tex]
8. [tex]\(3 = 3\)[/tex]
[tex]\[ 6561 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 = (3^4)^2 = (81)^2 \][/tex]
Por lo tanto:
[tex]\[ \sqrt{6561} = 81 \][/tex]
e. [tex]\(\sqrt{4096}\)[/tex]
1. [tex]\(4096 \div 2 = 2048\)[/tex]
2. [tex]\(2048 \div 2 = 1024\)[/tex]
3. [tex]\(1024 \div 2 = 512\)[/tex]
4. [tex]\(512 \div 2 = 256\)[/tex]
5. [tex]\(256 \div 2 = 128\)[/tex]
6. [tex]\(128 \div 2 = 64\)[/tex]
7. [tex]\(64 \div 2 = 32\)[/tex]
8. [tex]\(32 \div 2 = 16\)[/tex]
9. [tex]\(16 \div 2 = 8\)[/tex]
10. [tex]\(8 \div 2 = 4\)[/tex]
11. [tex]\(4 \div 2 = 2\)[/tex]
12. [tex]\(2 \div 2 = 1\)[/tex]
[tex]\[ 4096 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = (2^6)^2 = (64)^2 \][/tex]
Por lo tanto:
[tex]\[ \sqrt{4096} = 64 \][/tex]
f. [tex]\(\sqrt{625}\)[/tex]
1. [tex]\(625 \div 5 = 125\)[/tex]
2. [tex]\(125 \div 5 = 25\)[/tex]
3. [tex]\(25 = 5 \times 5\)[/tex]
[tex]\[ 625 = 5 \times 5 \times 5 \times 5 = (5^2)^2 = (25)^2 \][/tex]
Por lo tanto:
[tex]\[ \sqrt{625} = 25 \][/tex]
g. [tex]\(\sqrt{729}\)[/tex]
1. [tex]\(729 \div 3 = 243\)[/tex]
2. [tex]\(243 \div 3 = 81\)[/tex]
3. [tex]\(81 \div 3 = 27\)[/tex]
4. [tex]\(27 \div 3 = 9\)[/tex]
5. [tex]\(9 \div 3 = 3\)[/tex]
6. [tex]\(3 = 3\)[/tex]
[tex]\[ 729 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 = (3^3)^2 = (27)^2 \][/tex]
Por lo tanto:
[tex]\[ \sqrt{729} = 27 \][/tex]
h. [tex]\(\sqrt{256}\)[/tex]
1. [tex]\(256 \div 2 = 128\)[/tex]
2. [tex]\(128 \div 2 = 64\)[/tex]
3. [tex]\(64 \div 2 = 32\)[/tex]
4. [tex]\(32 \div 2 = 16\)[/tex]
5. [tex]\(16 \div 2 = 8\)[/tex]
6. [tex]\(8 \div 2 = 4\)[/tex]
7. [tex]\(4 \div 2 = 2\)[/tex]
8. [tex]\(2 = 2\)[/tex]
[tex]\[ 256 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = (2^4)^2 = (16)^2 \][/tex]
Por lo tanto:
[tex]\[ \sqrt{256} = 16 \][/tex]
Resumiendo:
a. [tex]\(\sqrt{1225} = 35\)[/tex]
b. [tex]\(\sqrt{1681} = 41\)[/tex]
c. [tex]\(\sqrt{400} = 20\)[/tex]
d. [tex]\(\sqrt{6561} = 81\)[/tex]
e. [tex]\(\sqrt{4096} = 64\)[/tex]
f. [tex]\(\sqrt{625} = 25\)[/tex]
g. [tex]\(\sqrt{729} = 27\)[/tex]
h. [tex]\(\sqrt{256} = 16\)[/tex]
a. [tex]\(\sqrt{1225}\)[/tex]
1. [tex]\(1225 \div 5 = 245\)[/tex]
2. [tex]\(245 \div 5 = 49\)[/tex]
3. [tex]\(49 = 7 \times 7\)[/tex]
[tex]\[ 1225 = 5 \times 5 \times 7 \times 7 = (5 \times 7)^2 \][/tex]
Por lo tanto:
[tex]\[ \sqrt{1225} = 35 \][/tex]
b. [tex]\(\sqrt{1681}\)[/tex]
1. [tex]\(1681 = 41 \times 41\)[/tex]
[tex]\[ 1681 = (41)^2 \][/tex]
Por lo tanto:
[tex]\[ \sqrt{1681} = 41 \][/tex]
c. [tex]\(\sqrt{400}\)[/tex]
1. [tex]\(400 \div 2 = 200\)[/tex]
2. [tex]\(200 \div 2 = 100\)[/tex]
3. [tex]\(100 \div 2 = 50\)[/tex]
4. [tex]\(50 \div 2 = 25\)[/tex]
5. [tex]\(25 = 5 \times 5\)[/tex]
[tex]\[ 400 = 2 \times 2 \times 2 \times 2 \times 5 \times 5 = (2 \times 2 \times 5)^2 = (20)^2 \][/tex]
Por lo tanto:
[tex]\[ \sqrt{400} = 20 \][/tex]
d. [tex]\(\sqrt{6561}\)[/tex]
1. [tex]\(6561 \div 3 = 2187\)[/tex]
2. [tex]\(2187 \div 3 = 729\)[/tex]
3. [tex]\(729 \div 3 = 243\)[/tex]
4. [tex]\(243 \div 3 = 81\)[/tex]
5. [tex]\(81 \div 3 = 27\)[/tex]
6. [tex]\(27 \div 3 = 9\)[/tex]
7. [tex]\(9 \div 3 = 3\)[/tex]
8. [tex]\(3 = 3\)[/tex]
[tex]\[ 6561 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 = (3^4)^2 = (81)^2 \][/tex]
Por lo tanto:
[tex]\[ \sqrt{6561} = 81 \][/tex]
e. [tex]\(\sqrt{4096}\)[/tex]
1. [tex]\(4096 \div 2 = 2048\)[/tex]
2. [tex]\(2048 \div 2 = 1024\)[/tex]
3. [tex]\(1024 \div 2 = 512\)[/tex]
4. [tex]\(512 \div 2 = 256\)[/tex]
5. [tex]\(256 \div 2 = 128\)[/tex]
6. [tex]\(128 \div 2 = 64\)[/tex]
7. [tex]\(64 \div 2 = 32\)[/tex]
8. [tex]\(32 \div 2 = 16\)[/tex]
9. [tex]\(16 \div 2 = 8\)[/tex]
10. [tex]\(8 \div 2 = 4\)[/tex]
11. [tex]\(4 \div 2 = 2\)[/tex]
12. [tex]\(2 \div 2 = 1\)[/tex]
[tex]\[ 4096 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = (2^6)^2 = (64)^2 \][/tex]
Por lo tanto:
[tex]\[ \sqrt{4096} = 64 \][/tex]
f. [tex]\(\sqrt{625}\)[/tex]
1. [tex]\(625 \div 5 = 125\)[/tex]
2. [tex]\(125 \div 5 = 25\)[/tex]
3. [tex]\(25 = 5 \times 5\)[/tex]
[tex]\[ 625 = 5 \times 5 \times 5 \times 5 = (5^2)^2 = (25)^2 \][/tex]
Por lo tanto:
[tex]\[ \sqrt{625} = 25 \][/tex]
g. [tex]\(\sqrt{729}\)[/tex]
1. [tex]\(729 \div 3 = 243\)[/tex]
2. [tex]\(243 \div 3 = 81\)[/tex]
3. [tex]\(81 \div 3 = 27\)[/tex]
4. [tex]\(27 \div 3 = 9\)[/tex]
5. [tex]\(9 \div 3 = 3\)[/tex]
6. [tex]\(3 = 3\)[/tex]
[tex]\[ 729 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 = (3^3)^2 = (27)^2 \][/tex]
Por lo tanto:
[tex]\[ \sqrt{729} = 27 \][/tex]
h. [tex]\(\sqrt{256}\)[/tex]
1. [tex]\(256 \div 2 = 128\)[/tex]
2. [tex]\(128 \div 2 = 64\)[/tex]
3. [tex]\(64 \div 2 = 32\)[/tex]
4. [tex]\(32 \div 2 = 16\)[/tex]
5. [tex]\(16 \div 2 = 8\)[/tex]
6. [tex]\(8 \div 2 = 4\)[/tex]
7. [tex]\(4 \div 2 = 2\)[/tex]
8. [tex]\(2 = 2\)[/tex]
[tex]\[ 256 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = (2^4)^2 = (16)^2 \][/tex]
Por lo tanto:
[tex]\[ \sqrt{256} = 16 \][/tex]
Resumiendo:
a. [tex]\(\sqrt{1225} = 35\)[/tex]
b. [tex]\(\sqrt{1681} = 41\)[/tex]
c. [tex]\(\sqrt{400} = 20\)[/tex]
d. [tex]\(\sqrt{6561} = 81\)[/tex]
e. [tex]\(\sqrt{4096} = 64\)[/tex]
f. [tex]\(\sqrt{625} = 25\)[/tex]
g. [tex]\(\sqrt{729} = 27\)[/tex]
h. [tex]\(\sqrt{256} = 16\)[/tex]
