$\sqrt{12345678987654321}$=?
$12345678987654321$=
$10^8$+$111 \times 10^7$+$11111 \times 10^6$+$1111111 \times 10^5$+$111111111 \times 10^4$+$11111111111 \times 10^3$+$1111111111111 \times 10^2$+$111111111111111 \times 10$+$11111111111111111$=
$\large\frac{1}{9}$$[(10-1)\times 10^8+(10^3-1)\times 10^7+(10^5-1)\times 10^6+(10^7-1)\times 10^5+(10^9-1)\times 10^4+(10^{11}-1)\times 10^3$
$+(10^{13}-1)\times 10^2+(10^{15}-1)\times 10+(10^{17}-1)]$=
$\large\frac{1}{9}$$[10^9(1+10+10^2+.....+10^7+10^8)-(10^8+10^7+10^6+....+10+1)]$=
$\large\frac{1}{9}$$[(10^9-1)(1+10+10^2+.....+10^7+10^8)]$=
$\large\frac{1}{9}$$[(10^9-1)\large\frac{1-10^9}{1-10}]$=$\large(\frac{10^9-1}{9})^2$=$111111111^2$
所以$\sqrt{12345678987654321}$=$111111111$
$111111111\times 111111111$=$111111111\times (1+10+10^2+10^3+.....+10^8)$
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