help!why!

since
(10a+5)^2
= 100a^2 + 100a+25
= 100a(a+1) +25

y i do this ?

example :
15 ^2 = 1 x (1 + 1) x 100 + 25 = 225 (15^2 means 15 x 15)

35^2 = 3 x (3 + 1)x 100 +25 = 1225 (35^2means 35 x 35)

so 25 never change for the last two digit bcoz only a will change in this formula and after that just very simpe for a<11

for the second one i just do 3 x(3 +1) then is 12 and then i just put 25 behind
= 1225

(try others if you don t believe)
(remember this only works for the number which the last digit is 5 and only for square)

now i can say 5 x 5 x 5 x 5 x 5 x 5 x 5
= 25 x 25 x 25 x 25 / 5
= 625 x 625 / 5

again i can do a formula :

(100a+10b+c)^2 (10b+c = 25 here)
=(100a +25 )^2
=10000a^2 +5000a+(25)^2 (we know that 25^2 = 625 from above)
=10000a(a+0.5) +625

again you can see if there have something25 ^2 appear, the last three digit will not change, there,s always 625

so
425^2 = 10000(4 x 4.5 )+625 = 180625 ( 4 x 4.5 = 4 x 4 + 4 x 0.5 = 16 + 2 = 18 )

525^2 = 10000(5 x 5.5 )+625 = 275625 (5 x 5.5 = 5 x 5 + 5 x 0.5 = 25+2.5 = 27.5)

and 625^2 = 10000(6 x 6.5 ) +625 = 390625

and now i can write :
5 x 5 x 5 x 5 x 5 x 5 x 5
= 625 x 625
= 390625 / 5
=390625x 2 / 10 (1/ 5 = 2 / 10)
=781250/ 10
=78125

all of the above i just using some shortcut

otherwise you have to do 5 x 5 x 5 x 5 x 5 x 5 x 5 yourself or use caculator

2007-11-27 07:15:36 補充：
anothershort cut :(10 +a )(10 + b)=100+10(a +b)+ab=10(10+a+b) +abtry this:11 x 12 = ( 11+2 )x 10 + 1 x 2 = 130 + 2 = 132 (11= 10 +a here , and b = 2 )19 x 18 = ( 19+8 )x 10 + 9 x 8 = 270 + 72 =342(this can only use between 10 and 19)

2007-11-27 07:22:38 補充：
y i do this again?combie the first and the third formula , i can easy say (115x 115 ) = (115)^2 = 13225(185x 185 ) = (185)^2 = 34425if you are asking something like that, hope this will helpful, otherwise better use your caculator

5*5*5*5*5*5*5
=25*5*5*5*5*5
=125*5*5*5*5
=625*5*5*5
=3125*5*5
=15625*5
=78125

78125

=78125

ok?