а) 216 = 6³ = 2³ · 3³;
162 = 2 · 81 = 2 · 3⁴;
144 = 12² = 2⁴ · 3²;
512 = 2 · 256 = 2² · 128 = 2³ · 64 = 2³ · 2⁶ = 2⁹;
675 = 5 · 135 = 5² · 27 = 3³ · 5²;
1024 = 512 · 2 = 2⁹ · 2 = 2¹⁰;
б) 60 = 4 · 15 = 2² · 3 · 5;
180 = 60 · 3 = 2² · 3² · 5;
220 = 22 · 10 = 2² · 5 · 11;
350 = 10 · 35 = 2 · 5² · 7;
1200 = 12 · 100 = 2² · 3 · 4 · 25 = 2⁴ · 3 · 5²;
8000 = 8 · 1000 = 2³ · 8 · 125 = 2³ · 2³ · 5³ = 2⁶ · 5³;
в) 11 = 11;
1001 = 7 · 11 · 13;
1225 = 5 · 245 = 5² · 49 = 5² · 7²;
21780 = 2 · 10890 = 2² · 5445 = 2² · 5 · 1089 = 2² · 5 · 9 · 121 = 2² · 3² · · 5 · 11²;
45630 = 2 · 22815 = 2 · 5 · 4563 = 2 · 5 · 9 · 507 = 2 · 3² · 5 · 3 · 169 =
= 2 · 3³ · 5 · 13².
1/3х + 1/5х + 1/6х = 21/40
1*10/3*10х + 1*6/5*6х + 1*5/6*5х = 21/40
10/30х + 6/30х + 5/30х = 21/40
21/30х = 21/40
х = 21/40 : 21/30
х = 21/40 * 30/21
х = 21*30/40*21
х = 30/40
х = 3/4
проверка:
1/3 * 3/4 + 1/5 * 3/4 + 1/6 * 3/4 = 21/40
3/12 + 3/20 + 3/24 = 21/40
3*10/12*10 + 3*6/20*6 + 3*5/24*5 = 21/40
30/120 + 18/120 + 15/120 = 21/40
63/120 = 21/40
21/40 = 21/40
2)
1/4х + 1/6х + 1/8х = 39/56
1*6/4*6х + 1*4/6*4х + 1*3/8*3х = 39/56
6/24х + 4/24х + 3/24х = 39/56
13/24х = 39/56
х = 39/56 : 13/24
х = 39/56 * 24/13
х = 39*24/56*13
х = 3*24/56
х = 72/56
х = 72:2/56:2
х = 36/28
х = 36:4/28:4
х = 9/7 или 1 2/7
проверка:
1/4 * 9/7 + 1/6 * 9/7 + 1/8 * 9/7 = 39/56
9/28 + 9/42 + 9/56 = 39/56
9*6/28*6 + 9*4/42*4 + 9*3/56*3 = 39/56
54/168 + 36/168 + 27/168 = 39/56
117/168 = 39/56
39/56 = 39/56