6 10x10=100
14x14 = 10x10+10+(11+12+13)x2+14
= 10x10+10+ (10x3+1+2+3)x2+14
=100+10+(30+6)x2+14
=100+10+72+14=100+72+24=196
0r14x14 = 10x10+(11+12+13+14)x2-4
= 10x10+ (10x4+1+2+3+4)x2- 4
=100+(40+10)x2-4=100+100-4
=200-4=196
25x25= 20x20+20+(21+22+23+24)x2+5x5
=400+20+(20x4+1+2+3+4)x2+25
=400+(21+22+23+24+25)x2-5
=400+(20x5+1+2+3+4+5)x2-5
=400+230-5=625
84x84 =80x80 +(80+81+82+83)x2+4x4
=6400+(80x4+1+2+3)x2+4x4
84x84=80x80+(81+82+83+84)x2-4
=6400+(80x4+10)x2-4
=6400+660-4
=7060-4
=7056
84x84=80x80+(80+81+82+83)x2+4
=6400+(80x4+6)x2+4
=6400+326x2+4= 6400+652+4
=7052+4
=7056
98x98= 90x90+2(90+91+92+93+94+95+96+97)+8=8100+2(720+28)+8=8100+1496+8=9604
27x27=20x20+2(20+21+22+23+24+25+26)+7=400+2(20x7+21)+7=400+322+7=729
123x123=120x120+2(120+121+122)+3=14400+2(120x3+1+2)+3=14400+2(360+3)+3=14400+2x363
+3=14400+726+3=15129
8x8=7x7+7x2+1
9x9=8x8+8x2+1
12x12=11x11+11x2+1
13x13=12x12+12x2+1
1x1=1 ; 1+ 1x2+1 = 2x2 ; 2x2+2x2+1=3x3 ; 3x3+3x2+1 =4x4
9x9=81= 8x8 +8+9=8x9+9
10x10=9x9+9+10 = 9x10+10
15x15=14x15+15
31x31=30x30+31x10+1x1-300 = 900+310+1-300=961
32x32=30x30+32x10+2x2-200 =900+320+4-200=1024
34x34=30x30+34x10+4x4-100=900+340+16-100=1256-100=1156
35x35=30x30+35x10+25-50=900+350+25-50=1225
36x36=30x30+36x10+6x6 = 900+360+36=1296=35x35+35+36=1225+71=1296
37x37=30x30+37x10+7x7+50=900+370+49+50=1369=36x36+36+37=1296+73=1369
38x38=30x30+38x10+8x8+100 = 900+380+64+100=1344+100=1444=1369+37+38=1369+75=1444
39x39=30x30+39x10+9x9+150 = 900+390+81+150 =1371+150=1521=1444+38+39=1444+77=1521
40x40=30x30+40x10+10x10+200=900+400+100+200=1600
93x93=93x10+2(83+73+63+53+43+33+23+13+3)x10-3x(10-3)=930+20x387-3x7=930+7740-21=8670-21
3x3x3=3x2+1+ 7+2+9(3x3)+2 =27
7x7x7 = 7x6+1 +7x6+1+2+47+49(7x7)+51+53+55(7x8-1)=
38x38=38x(40-2)=38x40-38x2
39x39=39x(40-1)=39x40-39x1
NXN=NX10+(N-10)X10+(N-10)2
15X15=15X10+2(15-10)+5X5
17X17=17X10+2(17-10)+7X7
27X27=2(27X10+7X10)+7X7
=2(27+7)X10)+7X7
29X29=2(29X10+9X10)+9X9
=2 (29+9)X10)+9X9
=2(38X10)+9X9=760+81=841
37X37=37X10+2(37-10)+27X27=370+270+729
0R =2(37X10+7X10)+7X7
=2(37+7)X10+7X7
123X123=120X120+2(120+121+122)+3=14400+726+3=15129
256X256=250X250+2(250+251+252+253+254+255)+6=62500+2(250X6+1+2+3+4+5)+6=62500+2(1500+15)+6=62500+3000+30+6=65536
By addition
1 67x67=60x10+2(57+47+37+M27+17+7)x10 +7x7=600+20x192+49=600+3840+7x7=4440+49=4489
In which 192 =150x10+6x7
2 Or67x67 =2(67+57+47+37+27+17)x10+7x7-600 +7x7=20x252+49-600=5040+49-600=5089-600=4489
For 252 = 6x7+210[= 10x7x(10-7)]
3method 67x67=67x10+2(57+47+37+27+17+7)x10-7x(10-7)=670+3840-21=4489
4 672 =2X (67+57+47+37+27+17+7) - 670-7X(10-7)=2X2590-670-7X3=5180-670-21=4510-21=4489
Substraction factor
93x93=93x10+2(83+73+63+53+43+33+23+13+3)x10-3x(10-3)=930+20x387-3x7=930+7740-21=8670-21
94x94=94x10+2(84+74+64+54+44+34+24+14+4)x10-4x(10-4)=940+20x396-4x6=940+7920-24=8860-24
98x98=98x10+2(88+78+68+58+48+38+28+18+8)x10-8x(10-8)=980+20x432-4x6=980+7740-16=9620-16
97x97=90x10+2(87+77+67+57+47+37+27+17+7)x10+7x7)=930+20x(360+9x7)+7x7=900+20x423+49=900+8460+49=9409
2 OR =97X10+8460-7X[10-7]=970+8460-21=9430-21=9409
3 OR=2 (97+87+77+67+57+47+37+27+17)x10+7x7-900=20X[450+ 9X7]+7X7-900=20X513+49-900=10260+49-900=9409
1 9422=9420+2x10(932+922+912+---------------------+12+2)-2x(10-2)=920+20(47x934)-2x(10-2)=9420+877960-16=887364 for 47x934=94/2[2x932+(94-1)x-10] A.P. taking place value10
2 9422=94200+ 2(842+742+642+---------------242+142+42)=94200+2x3978x100-42x(100-42)
=94200+795600-42x58=889800-2436=889800- 2436=887364 Taking place value 100 due to three digit number942
3 9422= + 2(942+842+742+642+---------------242+142+42) -42x(100-42) -94200 = -94200+2x3978x100-42x(100-42)
=2X4920X100-42X58-94200=984000-94200-2436 =889800-2436=887364
1 39422 = 3942000+2x(2942+1942+942)x1000=3942000+2(5700+126)x1000=3942000+2(5826x1000)-942x(1000-942)=3942000+11652000)-942x(1000-942) = 15594000-54636=15539364 Taking place value 1000 due to four digit number3942.
STATEMENT
1 SQUARE OF ANY ONE DIGIT NUMBER IS TWO TIMES OF SUMMATION OVER THE NUMBER WITH
ONE DECREASING MINUS PRODUCT OF DIGIT AND WITH IT’S SUBSTRATION FROM TEN .
9X9=2(9+8+7+6+5+4+3+2+1) -9X(10-9)=2X45 -9=81
2 SQUARE OF ANY TWO DIGIT NUMBER IS TWENTY TIMES OF SUMMATION OVER THE NUMBER WITH
TEN DECREASING MINUS PRODUCT OF DIGIT OF UNITPLACE AND WITH IT’S SUBSTRATION FROM TEN AND TENTIMES OF THAT NUMBER TOO .
1 24X24 =2(24+14+4)X10- 4 X (10-4)-240 =2X42X10-4X6 -240=840-240-24=576
2 672 =2X (67+57+47+37+27+17+7) - 670-7X(10-7)=2X2590-670-7X3=5180-670-21=4510-21=4489
3 SQUARE OF ANY THREE DIGIT NUMBER IS TWO HUNDREDS TIMES OF SUMMATION OVER THE NUMBER WITH
HUNDRED DECREASING MINUS PRODUCT OF DIGIT OF TENS &UNITPLACE AND WITH IT’S SUBSTRATION FROM HUNDREDS ANDIT IS WITH HUNDRED TIME OF THAT NUMBER TOO .
3 9422= + 2(942+842+742+642+---------------242+142+42) -42x(100-42) -94200 = -94200+2x3978x100-42x(100-42)
=2X4920X100-42X58-94200=984000-94200-2436 =889800-2436=887364
4 SQUARE OF ANY FOUR DIGIT NUMBER IS TWO THOUSANDS TIMES OF SUMMATION OVER THE NUMBER WITH
THOUSANDS DECREASING MINUS PRODUCT OF DIGIT OF HUNDREDS, TENS &UNITPLACE AND WITH IT’S SUBSTRATION FROM THOUSANDS AND IT IS WITH THOUSANDS TIME OF THAT NUMBER TOO .
NOTE –SIMILARLY IT CAN BE IMPLIMENTED FOR FIVE,SIX,SEVEN,EIGHT,NINE DIGITS AND MORE ON WARDS DIGITS REPLACING TEN THOUSANDS,LACS,TENLAKHS---S ON.
SHEET 3
1+58+360
1+59
1+60
1+61
1+62
1+63
1+64
=360+59+60+61+62+63+64=729
CUBE OFCONSECUTIVE NUMBERS
DIFFERENCEOF CUBESOF CONSECUTIVE
6X6X6=
1X1X1
1
7
12
6X1
6
2X2X2
8
19
18
(6-1)X6
30
3X3X3
27
37
24
(6-2)X12
48
4
64
61
30
[6-3]X18
54
5
125
91
36
[6-4]X24
48
6
216
127
42
[6-5]X30
30
7
343
169
48
216
8
512
217
54
9
729
217
60
10
1000
271
66
11
1331
331
72
12
1728
397
78
N3= NX 0+[[N-1]X n[N -2]X n+[N-3]X n
+[N-4]X n+[N-1]X5 n Where N is base number and n is power number,here n=3
7X7X7
NXNXN
7X1
7
NX1
N
[7-1]X6
36
[N-1]X6
6N-6
[7-2]X12
60
[N-2]X12
12N-24
[7-3]X18
72
[N-3]X18
18N-54
[7-4]X24]
72
[N-4]X24]
24N-96
[7-5]X30
60
[N-5]X30
30N-150
[7-6]X36
36
343
[N-[N-1].X6[N-1]
6N-6
97N-336
fourth POWER of
1
1
1
1
1
1
2
16
15
14
13
12
3
81
65
50
36
23
4
256
175
110
60
24
5
625
369
194
84
24
6
1296
671
302
108
24
7
2401
1105
434
132
24
8
4096
1695
9
6561
2465
10
10000
3439
11
14641
4641
12
20736
13
28561
NOTE—DUE TO POWER FOUR REPEATION START FROM
FORTH TERM IN LOST COLOUMN
5 thpower
1
1
1
1
1
1 1
2
32
31
30
29
28 27
3
243
211
180
150
121 93
4
1024
781
570
390
240 119
5
3125
2101
1320
750
360 120
6
7776
4651
2550
1230
480120
7
16807
9031
4380
1830
600 120
8
32768
9
59049
10
100000
NOTE—DUE TO POWER FIFTH REPEATION OF FACTORIAL 5=120 START FROM
FIFTH TERM IN LOST FIFTH COLOUMN
25 =1+1+1+1+1+27=32
35=1+27+93+1+28+30+31+32=243
n
6thPOWER
1
2
3
4
5
6
7
8
9
10
1
64
729
4096
15625
46656
117649
262144
531441
1000000
SUBTRACTS1
63
665
3367
11529
31031
70993
144495
1
62
602
2702
8162
19502
39962
73502
1
61
540
2100
5460
11340
20460
33540
1
60
479
1560
3360
5880
9120
13080
1
59
419
1081
1800
2520
3240
3960
1
58
360
662
719
720
720
720
ADD
1+58+360+662+719=
360+720+720
NOTE—DUE TO POWER SIXTH REPEATION OF FACTORIAL 6=720 START FROM
SISTH TERM IN LOST FIFTH
ROW
1
2
3
4
5
6
7
8
9
10
1
64
729
4096
15625
46656
117649
262144
531441
1000000
665
3367
11529
31031
70993
144495
2702
8162
19502
39962
73502
5460
11340
20460
33540
3360
5880
9120
13080
1800
2520
3240
3960
720
720
720
36=1+58+360
+60+61+62+63+64=729
7
1
2
3
4
5
6
7
8
9
10
1
128
2187
16384
78124
299936
823543
2097152
4782969
10000000
1
127
2059
14197
61740
201812
543607
1273609
2685817
5217031
1
126
1932
12138
47543
140072
341789
730008
1412208
2531213
10206
35605
92529
201717
388219
1339200
1119005
22399
56924
108188
186403
950981
220095
DIFFERENCES nth powers of consecutive numbers and again and again goes up to nth columns or row and repeated number [factorial n ] from nth term of nth columns row . And from reversing this process as sum or differences the value nth power of any number can be obtained without using operation of multiplication.
6 10x10=100
14x14 = 10x10+10+(11+12+13)x2+14
= 10x10+10+ (10x3+1+2+3)x2+14
=100+10+(30+6)x2+14
=100+10+72+14=100+72+24=196
0r14x14 = 10x10+(11+12+13+14)x2-4
= 10x10+ (10x4+1+2+3+4)x2- 4
=100+(40+10)x2-4=100+100-4
=200-4=196
25x25= 20x20+20+(21+22+23+24)x2+5x5
=400+20+(20x4+1+2+3+4)x2+25
=400+(21+22+23+24+25)x2-5
=400+(20x5+1+2+3+4+5)x2-5
=400+230-5=625
84x84 =80x80 +(80+81+82+83)x2+4x4
=6400+(80x4+1+2+3)x2+4x4
84x84=80x80+(81+82+83+84)x2-4
=6400+(80x4+10)x2-4
=6400+660-4
=7060-4
=7056
84x84=80x80+(80+81+82+83)x2+4
=6400+(80x4+6)x2+4
=6400+326x2+4= 6400+652+4
=7052+4
=7056
98x98= 90x90+2(90+91+92+93+94+95+96+97)+8=8100+2(720+28)+8=8100+1496+8=9604
27x27=20x20+2(20+21+22+23+24+25+26)+7=400+2(20x7+21)+7=400+322+7=729
123x123=120x120+2(120+121+122)+3=14400+2(120x3+1+2)+3=14400+2(360+3)+3=14400+2x363
+3=14400+726+3=15129
8x8=7x7+7x2+1
9x9=8x8+8x2+1
12x12=11x11+11x2+1
13x13=12x12+12x2+1
1x1=1 ; 1+ 1x2+1 = 2x2 ; 2x2+2x2+1=3x3 ; 3x3+3x2+1 =4x4
9x9=81= 8x8 +8+9=8x9+9
10x10=9x9+9+10 = 9x10+10
15x15=14x15+15
31x31=30x30+31x10+1x1-300 = 900+310+1-300=961
32x32=30x30+32x10+2x2-200 =900+320+4-200=1024
34x34=30x30+34x10+4x4-100=900+340+16-100=1256-100=1156
35x35=30x30+35x10+25-50=900+350+25-50=1225
36x36=30x30+36x10+6x6 = 900+360+36=1296=35x35+35+36=1225+71=1296
37x37=30x30+37x10+7x7+50=900+370+49+50=1369=36x36+36+37=1296+73=1369
38x38=30x30+38x10+8x8+100 = 900+380+64+100=1344+100=1444=1369+37+38=1369+75=1444
39x39=30x30+39x10+9x9+150 = 900+390+81+150 =1371+150=1521=1444+38+39=1444+77=1521
40x40=30x30+40x10+10x10+200=900+400+100+200=1600
93x93=93x10+2(83+73+63+53+43+33+23+13+3)x10-3x(10-3)=930+20x387-3x7=930+7740-21=8670-21
3x3x3=3x2+1+ 7+2+9(3x3)+2 =27
7x7x7 = 7x6+1 +7x6+1+2+47+49(7x7)+51+53+55(7x8-1)=
38x38=38x(40-2)=38x40-38x2
39x39=39x(40-1)=39x40-39x1
NXN=NX10+(N-10)X10+(N-10)2
15X15=15X10+2(15-10)+5X5
17X17=17X10+2(17-10)+7X7
27X27=2(27X10+7X10)+7X7
=2(27+7)X10)+7X7
29X29=2(29X10+9X10)+9X9
=2 (29+9)X10)+9X9
=2(38X10)+9X9=760+81=841
37X37=37X10+2(37-10)+27X27=370+270+729
0R =2(37X10+7X10)+7X7
=2(37+7)X10+7X7
123X123=120X120+2(120+121+122)+3=14400+726+3=15129
256X256=250X250+2(250+251+252+253+254+255)+6=62500+2(250X6+1+2+3+4+5)+6=62500+2(1500+15)+6=62500+3000+30+6=65536
By addition
67x67=60x10+2(57+47+37+M27+17+7)x10 +7x7=600+20x192+49=600+3840+7x7=4440+49=4489
In which 192 =150x10+6x7
2 Or67x67 =2(67+57+47+37+27+17)x10+7x7-600 +7x7=20x252+49-600=5040+49-600=5089-600=4489
For 252 = 6x7+210[= 10x7x(10-7)]
3method 67x67=67x10+2(57+47+37+27+17+7)x10-7x(10-7)=670+3840-21=4489
4 672 =2X (67+57+47+37+27+17+7) - 670-7X(10-7)=2X2590-670-7X3=5180-670-21=4510-21=4489
Substraction factor
93x93=93x10+2(83+73+63+53+43+33+23+13+3)x10-3x(10-3)=930+20x387-3x7=930+7740-21=8670-21
94x94=94x10+2(84+74+64+54+44+34+24+14+4)x10-4x(10-4)=940+20x396-4x6=940+7920-24=8860-24
98x98=98x10+2(88+78+68+58+48+38+28+18+8)x10-8x(10-8)=980+20x432-4x6=980+7740-16=9620-16
97x97=90x10+2(87+77+67+57+47+37+27+17+7)x10+7x7)=930+20x(360+9x7)+7x7=900+20x423+49=900+8460+49=9409
OR =97X10+8460-7X[10-7]=970+8460-21=9430-21=9409
OR=2 (97+87+77+67+57+47+37+27+17)x10+7x7-900=20X[450+ 9X7]+7X7-900=20X513+49-900=10260+49-900=9409
1 9422=9420+2x10(932+922+912+---------------------+12+2)-2x(10-2)=920+20(47x934)-2x(10-2)=9420+877960-16=887364 for 47x934=94/2[2x932+(94-1)x-10] A.P. taking place value10
2 9422=94200+ 2(842+742+642+---------------242+142+42)=94200+2x3978x100-42x(100-42)
=94200+795600-42x58=889800-2436=889800- 2436=887364 Taking place value 100 due to three digit number942
3 9422= + 2(942+842+742+642+---------------242+142+42) -42x(100-42) -94200 = -94200+2x3978x100-42x(100-42)
=2X4920X100-42X58-94200=984000-94200-2436 =889800-2436=887364
1 39422 = 3942000+2x(2942+1942+942)x1000=3942000+2(5700+126)x1000=3942000+2(5826x1000)-942x(1000-942)=3942000+11652000)-942x(1000-942) = 15594000-54636=15539364 Taking place value 1000 due to four digit number3942.
STATEMENT
1 SQUARE OF ANY ONE DIGIT NUMBER IS TWO TIMES OF SUMMATION OVER THE NUMBER WITH
ONE DECREASING MINUS PRODUCT OF DIGIT AND WITH IT’S SUBSTRATION FROM TEN .
9X9=2(9+8+7+6+5+4+3+2+1) -9X(10-9)=2X45 -9=81
2 SQUARE OF ANY TWO DIGIT NUMBER IS TWENTY TIMES OF SUMMATION OVER THE NUMBER WITH
TEN DECREASING MINUS PRODUCT OF DIGIT OF UNITPLACE AND WITH IT’S SUBSTRATION FROM TEN AND TENTIMES OF THAT NUMBER TOO .
1 24X24 =2(24+14+4)X10- 4 X (10-4)-240 =2X42X10-4X6 -240=840-240-24=576
2 672 =2X (67+57+47+37+27+17+7) - 670-7X(10-7)=2X2590-670-7X3=5180-670-21=4510-21=4489
3 SQUARE OF ANY THREE DIGIT NUMBER IS TWO HUNDREDS TIMES OF SUMMATION OVER THE NUMBER WITH
HUNDRED DECREASING MINUS PRODUCT OF DIGIT OF TENS &UNITPLACE AND WITH IT’S SUBSTRATION FROM HUNDREDS ANDIT IS WITH HUNDRED TIME OF THAT NUMBER TOO .
3 9422= + 2(942+842+742+642+---------------242+142+42) -42x(100-42) -94200 = -94200+2x3978x100-42x(100-42)
=2X4920X100-42X58-94200=984000-94200-2436 =889800-2436=887364
4 SQUARE OF ANY FOUR DIGIT NUMBER IS TWO THOUSANDS TIMES OF SUMMATION OVER THE NUMBER WITH
THOUSANDS DECREASING MINUS PRODUCT OF DIGIT OF HUNDREDS, TENS &UNITPLACE AND WITH IT’S SUBSTRATION FROM THOUSANDS AND IT IS WITH THOUSANDS TIME OF THAT NUMBER TOO .
NOTE –SIMILARLY IT CAN BE IMPLIMENTED FOR FIVE,SIX,SEVEN,EIGHT,NINE DIGITS AND MORE ON WARDS DIGITS REPLACING TEN THOUSANDS,LACS,TENLAKHS---S ON.
SHEET 3
1+58+360
1+59
1+60
1+61
1+62
1+63
1+64
=360+59+60+61+62+63+64=729
CUBE OFCONSECUTIVE NUMBERS DIFFERENCEOF CUBESOF CONSECUTIVE 6X6X6=
1X1X1 1 7 12 6X1 6
2X2X2 8 19 18 (6-1)X6 30
3X3X3 27 37 24 (6-2)X12 48
4 64 61 30 [6-3]X18 54
5 125 91 36 [6-4]X24 48
6 216 127 42 [6-5]X30 30
7 343 169 48 216
8 512 217 54
9 729 217 60
10 1000 271 66
11 1331 331 72
12 1728 397 78
N3= NX∟.0+[[N-1]X1.∟n[N -2]X2∟n+[N-3]X3∟n
+[N-4]X4∟n+[N-1]X5∟n Where N is base number and n is power number,here n=3
7X7X7 NXNXN
7X1 7 NX1 N
[7-1]X6 36 [N-1]X6 6N-6
[7-2]X12 60 [N-2]X12 12N-24
[7-3]X18 72 [N-3]X18 18N-54
[7-4]X24] 72 [N-4]X24] 24N-96
[7-5]X30 60 [N-5]X30 30N-150
[7-6]X36 36
343
[N-[N-1].X6[N-1] 6N-6
97N-336
fourth POWER of
1 1 1 1 1 1
2 16 15 14 13 12
3 81 65 50 36 23
4 256 175 110 60 24
5 625 369 194 84 24
6 1296 671 302 108 24
7 2401 1105 434 132 24
8 4096 1695
9 6561 2465
10 10000 3439
11 14641 4641
12 20736
13 28561
NOTE—DUE TO POWER FOUR REPEATION START FROM
FORTH TERM IN LOST COLOUMN
5 thpower
1 1 1 1 1 1 1
2 32 31 30 29 28 27
3 243 211 180 150 121 93
4 1024 781 570 390 240 119
5 3125 2101 1320 750 360 120
6 7776 4651 2550 1230 480120
7 16807 9031 4380 1830 600 120
8 32768
9 59049
10 100000
NOTE—DUE TO POWER FIFTH REPEATION OF FACTORIAL 5=120 START FROM
FIFTH TERM IN LOST FIFTH COLOUMN
25 =1+1+1+1+1+27=32
35=1+27+93+1+28+30+31+32=243
n
6thPOWER
1 2 3 4 5 6 7 8 9 10
1 64 729 4096 15625 46656 117649 262144 531441 1000000
SUBTRACTS1 63 665 3367 11529 31031 70993 144495
1 62 602 2702 8162 19502 39962 73502
1 61 540 2100 5460 11340 20460 33540
1 60 479 1560 3360 5880 9120 13080
1 59 419 1081 1800 2520 3240 3960
1 58 360 662 719 720 720 720
ADD 1+58+360+662+719= 360+720+720
NOTE—DUE TO POWER SIXTH REPEATION OF FACTORIAL 6=720 START FROM
SISTH TERM IN LOST FIFTH
ROW
1 2 3 4 5 6 7 8 9 10
1 64 729 4096 15625 46656 117649 262144 531441 1000000
665 3367 11529 31031 70993 144495
2702 8162 19502 39962 73502
5460 11340 20460 33540
3360 5880 9120 13080
1800 2520 3240 3960
720 720 720
36=1+58+360
+60+61+62+63+64=729
7
1 2 3 4 5 6 7 8 9 10
1 128 2187 16384 78124 299936 823543 2097152 4782969 10000000
1 127 2059 14197 61740 201812 543607 1273609 2685817 5217031
1 126 1932 12138 47543 140072 341789 730008 1412208 2531213
10206 35605 92529 201717 388219 1339200 1119005
22399 56924 108188 186403 950981 220095
DIFFERENCES nth powers of consecutive numbers and again and again goes up to nth columns or row and repeated number [factorial n ] from nth term of nth columns row . And from reversing this process as sum or differences the value nth power of any number can be obtained without using operation of multiplication.
6 10x10=100
14x14 = 10x10+10+(11+12+13)x2+14
= 10x10+10+ (10x3+1+2+3)x2+14
=100+10+(30+6)x2+14
=100+10+72+14=100+72+24=196
0r14x14 = 10x10+(11+12+13+14)x2-4
= 10x10+ (10x4+1+2+3+4)x2- 4
=100+(40+10)x2-4=100+100-4
=200-4=196
25x25= 20x20+20+(21+22+23+24)x2+5x5
=400+20+(20x4+1+2+3+4)x2+25
=400+(21+22+23+24+25)x2-5
=400+(20x5+1+2+3+4+5)x2-5
=400+230-5=625
84x84 =80x80 +(80+81+82+83)x2+4x4
=6400+(80x4+1+2+3)x2+4x4
84x84=80x80+(81+82+83+84)x2-4
=6400+(80x4+10)x2-4
=6400+660-4
=7060-4
=7056
84x84=80x80+(80+81+82+83)x2+4
=6400+(80x4+6)x2+4
=6400+326x2+4= 6400+652+4
=7052+4
=7056
98x98= 90x90+2(90+91+92+93+94+95+96+97)+8=8100+2(720+28)+8=8100+1496+8=9604
27x27=20x20+2(20+21+22+23+24+25+26)+7=400+2(20x7+21)+7=400+322+7=729
123x123=120x120+2(120+121+122)+3=14400+2(120x3+1+2)+3=14400+2(360+3)+3=14400+2x363
+3=14400+726+3=15129
8x8=7x7+7x2+1
9x9=8x8+8x2+1
12x12=11x11+11x2+1
13x13=12x12+12x2+1
1x1=1 ; 1+ 1x2+1 = 2x2 ; 2x2+2x2+1=3x3 ; 3x3+3x2+1 =4x4
9x9=81= 8x8 +8+9=8x9+9
10x10=9x9+9+10 = 9x10+10
15x15=14x15+15
31x31=30x30+31x10+1x1-300 = 900+310+1-300=961
32x32=30x30+32x10+2x2-200 =900+320+4-200=1024
34x34=30x30+34x10+4x4-100=900+340+16-100=1256-100=1156
35x35=30x30+35x10+25-50=900+350+25-50=1225
36x36=30x30+36x10+6x6 = 900+360+36=1296=35x35+35+36=1225+71=1296
37x37=30x30+37x10+7x7+50=900+370+49+50=1369=36x36+36+37=1296+73=1369
38x38=30x30+38x10+8x8+100 = 900+380+64+100=1344+100=1444=1369+37+38=1369+75=1444
39x39=30x30+39x10+9x9+150 = 900+390+81+150 =1371+150=1521=1444+38+39=1444+77=1521
40x40=30x30+40x10+10x10+200=900+400+100+200=1600
93x93=93x10+2(83+73+63+53+43+33+23+13+3)x10-3x(10-3)=930+20x387-3x7=930+7740-21=8670-21
3x3x3=3x2+1+ 7+2+9(3x3)+2 =27
7x7x7 = 7x6+1 +7x6+1+2+47+49(7x7)+51+53+55(7x8-1)=
38x38=38x(40-2)=38x40-38x2
39x39=39x(40-1)=39x40-39x1
NXN=NX10+(N-10)X10+(N-10)2
15X15=15X10+2(15-10)+5X5
17X17=17X10+2(17-10)+7X7
27X27=2(27X10+7X10)+7X7
=2(27+7)X10)+7X7
29X29=2(29X10+9X10)+9X9
=2 (29+9)X10)+9X9
=2(38X10)+9X9=760+81=841
37X37=37X10+2(37-10)+27X27=370+270+729
0R =2(37X10+7X10)+7X7
=2(37+7)X10+7X7
123X123=120X120+2(120+121+122)+3=14400+726+3=15129
256X256=250X250+2(250+251+252+253+254+255)+6=62500+2(250X6+1+2+3+4+5)+6=62500+2(1500+15)+6=62500+3000+30+6=65536
By addition
1 67x67=60x10+2(57+47+37+M27+17+7)x10 +7x7=600+20x192+49=600+3840+7x7=4440+49=4489
In which 192 =150x10+6x7
2 Or67x67 =2(67+57+47+37+27+17)x10+7x7-600 +7x7=20x252+49-600=5040+49-600=5089-600=4489
For 252 = 6x7+210[= 10x7x(10-7)]
3method 67x67=67x10+2(57+47+37+27+17+7)x10-7x(10-7)=670+3840-21=4489
4 672 =2X (67+57+47+37+27+17+7) - 670-7X(10-7)=2X2590-670-7X3=5180-670-21=4510-21=4489
Substraction factor
93x93=93x10+2(83+73+63+53+43+33+23+13+3)x10-3x(10-3)=930+20x387-3x7=930+7740-21=8670-21
94x94=94x10+2(84+74+64+54+44+34+24+14+4)x10-4x(10-4)=940+20x396-4x6=940+7920-24=8860-24
98x98=98x10+2(88+78+68+58+48+38+28+18+8)x10-8x(10-8)=980+20x432-4x6=980+7740-16=9620-16
97x97=90x10+2(87+77+67+57+47+37+27+17+7)x10+7x7)=930+20x(360+9x7)+7x7=900+20x423+49=900+8460+49=9409
2 OR =97X10+8460-7X[10-7]=970+8460-21=9430-21=9409
3 OR=2 (97+87+77+67+57+47+37+27+17)x10+7x7-900=20X[450+ 9X7]+7X7-900=20X513+49-900=10260+49-900=9409
1 9422=9420+2x10(932+922+912+---------------------+12+2)-2x(10-2)=920+20(47x934)-2x(10-2)=9420+877960-16=887364 for 47x934=94/2[2x932+(94-1)x-10] A.P. taking place value10
2 9422=94200+ 2(842+742+642+---------------242+142+42)=94200+2x3978x100-42x(100-42)
=94200+795600-42x58=889800-2436=889800- 2436=887364 Taking place value 100 due to three digit number942
3 9422= + 2(942+842+742+642+---------------242+142+42) -42x(100-42) -94200 = -94200+2x3978x100-42x(100-42)
=2X4920X100-42X58-94200=984000-94200-2436 =889800-2436=887364
1 39422 = 3942000+2x(2942+1942+942)x1000=3942000+2(5700+126)x1000=3942000+2(5826x1000)-942x(1000-942)=3942000+11652000)-942x(1000-942) = 15594000-54636=15539364 Taking place value 1000 due to four digit number3942.
STATEMENT
1 SQUARE OF ANY ONE DIGIT NUMBER IS TWO TIMES OF SUMMATION OVER THE NUMBER WITH
ONE DECREASING MINUS PRODUCT OF DIGIT AND WITH IT’S SUBSTRATION FROM TEN .
9X9=2(9+8+7+6+5+4+3+2+1) -9X(10-9)=2X45 -9=81
2 SQUARE OF ANY TWO DIGIT NUMBER IS TWENTY TIMES OF SUMMATION OVER THE NUMBER WITH
TEN DECREASING MINUS PRODUCT OF DIGIT OF UNITPLACE AND WITH IT’S SUBSTRATION FROM TEN AND TENTIMES OF THAT NUMBER TOO .
1 24X24 =2(24+14+4)X10- 4 X (10-4)-240 =2X42X10-4X6 -240=840-240-24=576
2 672 =2X (67+57+47+37+27+17+7) - 670-7X(10-7)=2X2590-670-7X3=5180-670-21=4510-21=4489
3 SQUARE OF ANY THREE DIGIT NUMBER IS TWO HUNDREDS TIMES OF SUMMATION OVER THE NUMBER WITH
HUNDRED DECREASING MINUS PRODUCT OF DIGIT OF TENS &UNITPLACE AND WITH IT’S SUBSTRATION FROM HUNDREDS ANDIT IS WITH HUNDRED TIME OF THAT NUMBER TOO .
3 9422= + 2(942+842+742+642+---------------242+142+42) -42x(100-42) -94200 = -94200+2x3978x100-42x(100-42)
=2X4920X100-42X58-94200=984000-94200-2436 =889800-2436=887364
4 SQUARE OF ANY FOUR DIGIT NUMBER IS TWO THOUSANDS TIMES OF SUMMATION OVER THE NUMBER WITH
THOUSANDS DECREASING MINUS PRODUCT OF DIGIT OF HUNDREDS, TENS &UNITPLACE AND WITH IT’S SUBSTRATION FROM THOUSANDS AND IT IS WITH THOUSANDS TIME OF THAT NUMBER TOO .
NOTE –SIMILARLY IT CAN BE IMPLIMENTED FOR FIVE,SIX,SEVEN,EIGHT,NINE DIGITS AND MORE ON WARDS DIGITS REPLACING TEN THOUSANDS,LACS,TENLAKHS---S ON.
SHEET 3
1+58+360
1+59
1+60
1+61
1+62
1+63
1+64
=360+59+60+61+62+63+64=729
CUBE OFCONSECUTIVE NUMBERS
DIFFERENCEOF CUBESOF CONSECUTIVE
6X6X6=
1X1X1
1
7
12
6X1
6
2X2X2
8
19
18
(6-1)X6
30
3X3X3
27
37
24
(6-2)X12
48
4
64
61
30
[6-3]X18
54
5
125
91
36
[6-4]X24
48
6
216
127
42
[6-5]X30
30
7
343
169
48
216
8
512
217
54
9
729
217
60
10
1000
271
66
11
1331
331
72
12
1728
397
78
N3= NX 0+[[N-1]X n[N -2]X n+[N-3]X n
+[N-4]X n+[N-1]X5 n Where N is base number and n is power number,here n=3
7X7X7
NXNXN
7X1
7
NX1
N
[7-1]X6
36
[N-1]X6
6N-6
[7-2]X12
60
[N-2]X12
12N-24
[7-3]X18
72
[N-3]X18
18N-54
[7-4]X24]
72
[N-4]X24]
24N-96
[7-5]X30
60
[N-5]X30
30N-150
[7-6]X36
36
343
[N-[N-1].X6[N-1]
6N-6
97N-336
fourth POWER of
1
1
1
1
1
1
2
16
15
14
13
12
3
81
65
50
36
23
4
256
175
110
60
24
5
625
369
194
84
24
6
1296
671
302
108
24
7
2401
1105
434
132
24
8
4096
1695
9
6561
2465
10
10000
3439
11
14641
4641
12
20736
13
28561
NOTE—DUE TO POWER FOUR REPEATION START FROM
FORTH TERM IN LOST COLOUMN
5 thpower
1
1
1
1
1
1 1
2
32
31
30
29
28 27
3
243
211
180
150
121 93
4
1024
781
570
390
240 119
5
3125
2101
1320
750
360 120
6
7776
4651
2550
1230
480120
7
16807
9031
4380
1830
600 120
8
32768
9
59049
10
100000
NOTE—DUE TO POWER FIFTH REPEATION OF FACTORIAL 5=120 START FROM
FIFTH TERM IN LOST FIFTH COLOUMN
25 =1+1+1+1+1+27=32
35=1+27+93+1+28+30+31+32=243
n
6thPOWER
1
2
3
4
5
6
7
8
9
10
1
64
729
4096
15625
46656
117649
262144
531441
1000000
SUBTRACTS1
63
665
3367
11529
31031
70993
144495
1
62
602
2702
8162
19502
39962
73502
1
61
540
2100
5460
11340
20460
33540
1
60
479
1560
3360
5880
9120
13080
1
59
419
1081
1800
2520
3240
3960
1
58
360
662
719
720
720
720
ADD
1+58+360+662+719=
360+720+720
NOTE—DUE TO POWER SIXTH REPEATION OF FACTORIAL 6=720 START FROM
SISTH TERM IN LOST FIFTH
ROW
1
2
3
4
5
6
7
8
9
10
1
64
729
4096
15625
46656
117649
262144
531441
1000000
665
3367
11529
31031
70993
144495
2702
8162
19502
39962
73502
5460
11340
20460
33540
3360
5880
9120
13080
1800
2520
3240
3960
720
720
720
36=1+58+360
+60+61+62+63+64=729
7
1
2
3
4
5
6
7
8
9
10
1
128
2187
16384
78124
299936
823543
2097152
4782969
10000000
1
127
2059
14197
61740
201812
543607
1273609
2685817
5217031
1
126
1932
12138
47543
140072
341789
730008
1412208
2531213
10206
35605
92529
201717
388219
1339200
1119005
22399
56924
108188
186403
950981
220095
DIFFERENCES nth powers of consecutive numbers and again and again goes up to nth columns or row and repeated number [factorial n ] from nth term of nth columns row . And from reversing this process as sum or differences the value nth power of any number can be obtained without using operation of multiplication.
6 10x10=100
14x14 = 10x10+10+(11+12+13)x2+14
= 10x10+10+ (10x3+1+2+3)x2+14
=100+10+(30+6)x2+14
=100+10+72+14=100+72+24=196
0r14x14 = 10x10+(11+12+13+14)x2-4
= 10x10+ (10x4+1+2+3+4)x2- 4
=100+(40+10)x2-4=100+100-4
=200-4=196
25x25= 20x20+20+(21+22+23+24)x2+5x5
=400+20+(20x4+1+2+3+4)x2+25
=400+(21+22+23+24+25)x2-5
=400+(20x5+1+2+3+4+5)x2-5
=400+230-5=625
84x84 =80x80 +(80+81+82+83)x2+4x4
=6400+(80x4+1+2+3)x2+4x4
84x84=80x80+(81+82+83+84)x2-4
=6400+(80x4+10)x2-4
=6400+660-4
=7060-4
=7056
84x84=80x80+(80+81+82+83)x2+4
=6400+(80x4+6)x2+4
=6400+326x2+4= 6400+652+4
=7052+4
=7056
98x98= 90x90+2(90+91+92+93+94+95+96+97)+8=8100+2(720+28)+8=8100+1496+8=9604
27x27=20x20+2(20+21+22+23+24+25+26)+7=400+2(20x7+21)+7=400+322+7=729
123x123=120x120+2(120+121+122)+3=14400+2(120x3+1+2)+3=14400+2(360+3)+3=14400+2x363
+3=14400+726+3=15129
8x8=7x7+7x2+1
9x9=8x8+8x2+1
12x12=11x11+11x2+1
13x13=12x12+12x2+1
1x1=1 ; 1+ 1x2+1 = 2x2 ; 2x2+2x2+1=3x3 ; 3x3+3x2+1 =4x4
9x9=81= 8x8 +8+9=8x9+9
10x10=9x9+9+10 = 9x10+10
15x15=14x15+15
31x31=30x30+31x10+1x1-300 = 900+310+1-300=961
32x32=30x30+32x10+2x2-200 =900+320+4-200=1024
34x34=30x30+34x10+4x4-100=900+340+16-100=1256-100=1156
35x35=30x30+35x10+25-50=900+350+25-50=1225
36x36=30x30+36x10+6x6 = 900+360+36=1296=35x35+35+36=1225+71=1296
37x37=30x30+37x10+7x7+50=900+370+49+50=1369=36x36+36+37=1296+73=1369
38x38=30x30+38x10+8x8+100 = 900+380+64+100=1344+100=1444=1369+37+38=1369+75=1444
39x39=30x30+39x10+9x9+150 = 900+390+81+150 =1371+150=1521=1444+38+39=1444+77=1521
40x40=30x30+40x10+10x10+200=900+400+100+200=1600
93x93=93x10+2(83+73+63+53+43+33+23+13+3)x10-3x(10-3)=930+20x387-3x7=930+7740-21=8670-21
3x3x3=3x2+1+ 7+2+9(3x3)+2 =27
7x7x7 = 7x6+1 +7x6+1+2+47+49(7x7)+51+53+55(7x8-1)=
38x38=38x(40-2)=38x40-38x2
39x39=39x(40-1)=39x40-39x1
NXN=NX10+(N-10)X10+(N-10)2
15X15=15X10+2(15-10)+5X5
17X17=17X10+2(17-10)+7X7
27X27=2(27X10+7X10)+7X7
=2(27+7)X10)+7X7
29X29=2(29X10+9X10)+9X9
=2 (29+9)X10)+9X9
=2(38X10)+9X9=760+81=841
37X37=37X10+2(37-10)+27X27=370+270+729
0R =2(37X10+7X10)+7X7
=2(37+7)X10+7X7
123X123=120X120+2(120+121+122)+3=14400+726+3=15129
256X256=250X250+2(250+251+252+253+254+255)+6=62500+2(250X6+1+2+3+4+5)+6=62500+2(1500+15)+6=62500+3000+30+6=65536
By addition
67x67=60x10+2(57+47+37+M27+17+7)x10 +7x7=600+20x192+49=600+3840+7x7=4440+49=4489
In which 192 =150x10+6x7
2 Or67x67 =2(67+57+47+37+27+17)x10+7x7-600 +7x7=20x252+49-600=5040+49-600=5089-600=4489
For 252 = 6x7+210[= 10x7x(10-7)]
3method 67x67=67x10+2(57+47+37+27+17+7)x10-7x(10-7)=670+3840-21=4489
4 672 =2X (67+57+47+37+27+17+7) - 670-7X(10-7)=2X2590-670-7X3=5180-670-21=4510-21=4489
Substraction factor
93x93=93x10+2(83+73+63+53+43+33+23+13+3)x10-3x(10-3)=930+20x387-3x7=930+7740-21=8670-21
94x94=94x10+2(84+74+64+54+44+34+24+14+4)x10-4x(10-4)=940+20x396-4x6=940+7920-24=8860-24
98x98=98x10+2(88+78+68+58+48+38+28+18+8)x10-8x(10-8)=980+20x432-4x6=980+7740-16=9620-16
97x97=90x10+2(87+77+67+57+47+37+27+17+7)x10+7x7)=930+20x(360+9x7)+7x7=900+20x423+49=900+8460+49=9409
OR =97X10+8460-7X[10-7]=970+8460-21=9430-21=9409
OR=2 (97+87+77+67+57+47+37+27+17)x10+7x7-900=20X[450+ 9X7]+7X7-900=20X513+49-900=10260+49-900=9409
1 9422=9420+2x10(932+922+912+---------------------+12+2)-2x(10-2)=920+20(47x934)-2x(10-2)=9420+877960-16=887364 for 47x934=94/2[2x932+(94-1)x-10] A.P. taking place value10
2 9422=94200+ 2(842+742+642+---------------242+142+42)=94200+2x3978x100-42x(100-42)
=94200+795600-42x58=889800-2436=889800- 2436=887364 Taking place value 100 due to three digit number942
3 9422= + 2(942+842+742+642+---------------242+142+42) -42x(100-42) -94200 = -94200+2x3978x100-42x(100-42)
=2X4920X100-42X58-94200=984000-94200-2436 =889800-2436=887364
1 39422 = 3942000+2x(2942+1942+942)x1000=3942000+2(5700+126)x1000=3942000+2(5826x1000)-942x(1000-942)=3942000+11652000)-942x(1000-942) = 15594000-54636=15539364 Taking place value 1000 due to four digit number3942.
STATEMENT
1 SQUARE OF ANY ONE DIGIT NUMBER IS TWO TIMES OF SUMMATION OVER THE NUMBER WITH
ONE DECREASING MINUS PRODUCT OF DIGIT AND WITH IT’S SUBSTRATION FROM TEN .
9X9=2(9+8+7+6+5+4+3+2+1) -9X(10-9)=2X45 -9=81
2 SQUARE OF ANY TWO DIGIT NUMBER IS TWENTY TIMES OF SUMMATION OVER THE NUMBER WITH
TEN DECREASING MINUS PRODUCT OF DIGIT OF UNITPLACE AND WITH IT’S SUBSTRATION FROM TEN AND TENTIMES OF THAT NUMBER TOO .
1 24X24 =2(24+14+4)X10- 4 X (10-4)-240 =2X42X10-4X6 -240=840-240-24=576
2 672 =2X (67+57+47+37+27+17+7) - 670-7X(10-7)=2X2590-670-7X3=5180-670-21=4510-21=4489
3 SQUARE OF ANY THREE DIGIT NUMBER IS TWO HUNDREDS TIMES OF SUMMATION OVER THE NUMBER WITH
HUNDRED DECREASING MINUS PRODUCT OF DIGIT OF TENS &UNITPLACE AND WITH IT’S SUBSTRATION FROM HUNDREDS ANDIT IS WITH HUNDRED TIME OF THAT NUMBER TOO .
3 9422= + 2(942+842+742+642+---------------242+142+42) -42x(100-42) -94200 = -94200+2x3978x100-42x(100-42)
=2X4920X100-42X58-94200=984000-94200-2436 =889800-2436=887364
4 SQUARE OF ANY FOUR DIGIT NUMBER IS TWO THOUSANDS TIMES OF SUMMATION OVER THE NUMBER WITH
THOUSANDS DECREASING MINUS PRODUCT OF DIGIT OF HUNDREDS, TENS &UNITPLACE AND WITH IT’S SUBSTRATION FROM THOUSANDS AND IT IS WITH THOUSANDS TIME OF THAT NUMBER TOO .
NOTE –SIMILARLY IT CAN BE IMPLIMENTED FOR FIVE,SIX,SEVEN,EIGHT,NINE DIGITS AND MORE ON WARDS DIGITS REPLACING TEN THOUSANDS,LACS,TENLAKHS---S ON.
SHEET 3
1+58+360
1+59
1+60
1+61
1+62
1+63
1+64
=360+59+60+61+62+63+64=729
CUBE OFCONSECUTIVE NUMBERS DIFFERENCEOF CUBESOF CONSECUTIVE 6X6X6=
1X1X1 1 7 12 6X1 6
2X2X2 8 19 18 (6-1)X6 30
3X3X3 27 37 24 (6-2)X12 48
4 64 61 30 [6-3]X18 54
5 125 91 36 [6-4]X24 48
6 216 127 42 [6-5]X30 30
7 343 169 48 216
8 512 217 54
9 729 217 60
10 1000 271 66
11 1331 331 72
12 1728 397 78
N3= NX∟.0+[[N-1]X1.∟n[N -2]X2∟n+[N-3]X3∟n
+[N-4]X4∟n+[N-1]X5∟n Where N is base number and n is power number,here n=3
7X7X7 NXNXN
7X1 7 NX1 N
[7-1]X6 36 [N-1]X6 6N-6
[7-2]X12 60 [N-2]X12 12N-24
[7-3]X18 72 [N-3]X18 18N-54
[7-4]X24] 72 [N-4]X24] 24N-96
[7-5]X30 60 [N-5]X30 30N-150
[7-6]X36 36
343
[N-[N-1].X6[N-1] 6N-6
97N-336
fourth POWER of
1 1 1 1 1 1
2 16 15 14 13 12
3 81 65 50 36 23
4 256 175 110 60 24
5 625 369 194 84 24
6 1296 671 302 108 24
7 2401 1105 434 132 24
8 4096 1695
9 6561 2465
10 10000 3439
11 14641 4641
12 20736
13 28561
NOTE—DUE TO POWER FOUR REPEATION START FROM
FORTH TERM IN LOST COLOUMN
5 thpower
1 1 1 1 1 1 1
2 32 31 30 29 28 27
3 243 211 180 150 121 93
4 1024 781 570 390 240 119
5 3125 2101 1320 750 360 120
6 7776 4651 2550 1230 480120
7 16807 9031 4380 1830 600 120
8 32768
9 59049
10 100000
NOTE—DUE TO POWER FIFTH REPEATION OF FACTORIAL 5=120 START FROM
FIFTH TERM IN LOST FIFTH COLOUMN
25 =1+1+1+1+1+27=32
35=1+27+93+1+28+30+31+32=243
n
6thPOWER
1 2 3 4 5 6 7 8 9 10
1 64 729 4096 15625 46656 117649 262144 531441 1000000
SUBTRACTS1 63 665 3367 11529 31031 70993 144495
1 62 602 2702 8162 19502 39962 73502
1 61 540 2100 5460 11340 20460 33540
1 60 479 1560 3360 5880 9120 13080
1 59 419 1081 1800 2520 3240 3960
1 58 360 662 719 720 720 720
ADD 1+58+360+662+719= 360+720+720
NOTE—DUE TO POWER SIXTH REPEATION OF FACTORIAL 6=720 START FROM
SISTH TERM IN LOST FIFTH
ROW
1 2 3 4 5 6 7 8 9 10
1 64 729 4096 15625 46656 117649 262144 531441 1000000
665 3367 11529 31031 70993 144495
2702 8162 19502 39962 73502
5460 11340 20460 33540
3360 5880 9120 13080
1800 2520 3240 3960
720 720 720
36=1+58+360
+60+61+62+63+64=729
7
1 2 3 4 5 6 7 8 9 10
1 128 2187 16384 78124 299936 823543 2097152 4782969 10000000
1 127 2059 14197 61740 201812 543607 1273609 2685817 5217031
1 126 1932 12138 47543 140072 341789 730008 1412208 2531213
10206 35605 92529 201717 388219 1339200 1119005
22399 56924 108188 186403 950981 220095
DIFFERENCES nth powers of consecutive numbers and again and again goes up to nth columns or row and repeated number [factorial n ] from nth term of nth columns row . And from reversing this process as sum or differences the value nth power of any number can be obtained without using operation of multiplication.
6 10x10=100
14x14 = 10x10+10+(11+12+13)x2+14
= 10x10+10+ (10x3+1+2+3)x2+14
=100+10+(30+6)x2+14
=100+10+72+14=100+72+24=196
0r14x14 = 10x10+(11+12+13+14)x2-4
= 10x10+ (10x4+1+2+3+4)x2- 4
=100+(40+10)x2-4=100+100-4
=200-4=196
25x25= 20x20+20+(21+22+23+24)x2+5x5
=400+20+(20x4+1+2+3+4)x2+25
=400+(21+22+23+24+25)x2-5
=400+(20x5+1+2+3+4+5)x2-5
=400+230-5=625
84x84 =80x80 +(80+81+82+83)x2+4x4
=6400+(80x4+1+2+3)x2+4x4
84x84=80x80+(81+82+83+84)x2-4
=6400+(80x4+10)x2-4
=6400+660-4
=7060-4
=7056
84x84=80x80+(80+81+82+83)x2+4
=6400+(80x4+6)x2+4
=6400+326x2+4= 6400+652+4
=7052+4
=7056
98x98= 90x90+2(90+91+92+93+94+95+96+97)+8=8100+2(720+28)+8=8100+1496+8=9604
27x27=20x20+2(20+21+22+23+24+25+26)+7=400+2(20x7+21)+7=400+322+7=729
123x123=120x120+2(120+121+122)+3=14400+2(120x3+1+2)+3=14400+2(360+3)+3=14400+2x363
+3=14400+726+3=15129
8x8=7x7+7x2+1
9x9=8x8+8x2+1
12x12=11x11+11x2+1
13x13=12x12+12x2+1
1x1=1 ; 1+ 1x2+1 = 2x2 ; 2x2+2x2+1=3x3 ; 3x3+3x2+1 =4x4
9x9=81= 8x8 +8+9=8x9+9
10x10=9x9+9+10 = 9x10+10
15x15=14x15+15
31x31=30x30+31x10+1x1-300 = 900+310+1-300=961
32x32=30x30+32x10+2x2-200 =900+320+4-200=1024
34x34=30x30+34x10+4x4-100=900+340+16-100=1256-100=1156
35x35=30x30+35x10+25-50=900+350+25-50=1225
36x36=30x30+36x10+6x6 = 900+360+36=1296=35x35+35+36=1225+71=1296
37x37=30x30+37x10+7x7+50=900+370+49+50=1369=36x36+36+37=1296+73=1369
38x38=30x30+38x10+8x8+100 = 900+380+64+100=1344+100=1444=1369+37+38=1369+75=1444
39x39=30x30+39x10+9x9+150 = 900+390+81+150 =1371+150=1521=1444+38+39=1444+77=1521
40x40=30x30+40x10+10x10+200=900+400+100+200=1600
93x93=93x10+2(83+73+63+53+43+33+23+13+3)x10-3x(10-3)=930+20x387-3x7=930+7740-21=8670-21
3x3x3=3x2+1+ 7+2+9(3x3)+2 =27
7x7x7 = 7x6+1 +7x6+1+2+47+49(7x7)+51+53+55(7x8-1)=
38x38=38x(40-2)=38x40-38x2
39x39=39x(40-1)=39x40-39x1
NXN=NX10+(N-10)X10+(N-10)2
15X15=15X10+2(15-10)+5X5
17X17=17X10+2(17-10)+7X7
27X27=2(27X10+7X10)+7X7
=2(27+7)X10)+7X7
29X29=2(29X10+9X10)+9X9
=2 (29+9)X10)+9X9
=2(38X10)+9X9=760+81=841
37X37=37X10+2(37-10)+27X27=370+270+729
0R =2(37X10+7X10)+7X7
=2(37+7)X10+7X7
123X123=120X120+2(120+121+122)+3=14400+726+3=15129
256X25
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