Try to solve this without a calculator

This is simple.

The denominator is

$1+2+3+4+5+6+7+8+9+8+7+6+5+4+3+2+1.$

Now group some terms,

$(1+2+3+4+5+6+7+8+9)+(8+7+6+5+4+3+2+1) = (45)+(36) = 81.$

Now , the fraction becomes

$\dfrac{999999 * 999999}{81} = \dfrac{999999*999999}{9*9} = 111111*111111 = \boxed{12345654321}.$

I did it like this: The sum of the numbers from 1 to 9 is 45.

The denominator becomes: 2 * (45) - 9 = 2 * (9 * 5) - 9 = 9 * (10 - 1) = 9 * 9.

Therefore the whole thing becomes: (999999 * 999999) / (9 * 9) = 111111 * 111111.

Investigating how to multiply numbers with digit series of ones:

11 * 11 = 121

111 * 111 = 12321

...

111111 * 111111 = 12345654321.

$\LARGE\frac{999999\times999999}{1+2+3+4+5+6+7+8+9+8+7+6+5+4+3+2+1}$

The denominator can be simplified as follows

$\underbrace{1+\overbrace{2+\underbrace{3+\overbrace{4+5}^9+6}_9+7}^9+8}_9+9+\underbrace{1+\overbrace{2+\underbrace{3+\overbrace{4+5}^9+6}_9+7}^9+8}_9$

$\LARGE=\frac{999999\times999999}{9+9+9+9+9+9+9+9+9}$ a summation of 9 {9's} which means 9×9

$\Large=\frac{999999\times999999}{9\times9}$

$\large=111111\times111111$

$\large=11111100000+1111110000+111111000+11111100+1111110+111111$

$\large\textbf{So the answer}=\color{#3D99F6}{\boxed{\color{cyan}{\boxed{\color{maroon}{12345654321}}}}}$

No calculator! The denominator is

$1+2+3+4+5+6+7+8+9+8+7+6+5+4+3+2+1 = 9 \times 9 = (10-1)(10-1)=(10-1)^2$ .

while the numerator can be rewritten as

$999999 \times 999999 = (10^6 -1)(10^6 -1)=(10^6 -1)^2$ .

Thus, the given expression is equal to

$\frac{(10^6 -1)^2}{(10-1)^2} =(\frac{10^6 -1}{10-1})^2$ .

Now, using the partial sum of geometric series,

$\frac{10^6 -1}{10-1} = \sum_{k=0}^5 10^k$

we have that

$(\frac{10^7 -1}{10-1})^2 = (\sum_{k=0}^5 10^k)^2 = 1 \times 10^{10} + 2\times 10^{9} +\dots+ 6\times 10^{5} + 5 \times 10^{4} + \dots + 1\times 10^{0}$

that is in base $10$ the number

$\boxed{12345654321}$ .

It's a combination of a few different tricks.

First, the denominator:

$1 + 2 + 3 + \ldots + 8 + 9 + 8 + \ldots + 3 + 2 + 1 = 2 ( 1 + 2 + 3 + \ldots + 8 ) + 9 = 2 \times 36 + 9 = 81$

Now, the numerator:

$999999 \times 999999 = 9 ( 111111 ) \times 9 ( 111111 ) = 81 ( 111111 \times 111111 )$

Putting it together:

$\frac { 81 ( 111111 \times 111111 ) }{ 81 } = 111111 \times 111111$

Finally, multiplying this is a simple enough trick: There are six 1's in each number, so the final answer will count up to 6 and then back down:

$111111 \times 111111 = 12345654321$

Okay.

1+2+3+4+5+6+7+8+9+8+7+6+5+4+3+2+1=81 (Which is the denominator)

81=9∗9

Simplifying the top we get 111111∗111111

Remembering,

$11^2$ =121

$111^2$ =12321

So, by Induction, $111111^2$ =12345654321

Without using calculator,

999999*999999/[9+9+9+9+9+9+9+9+9]

= 111111*999999/[1+1+1+1+1+1+1+1+1]

= 111111*111111/[1]

= 111111 + 1111110 + 11111100 + 111111000 + 1111110000 + 11111100000

= 12345654321

Denominator 1+2+3+4+5+6+7+8+9+8+7+6+5+4+3+2+1. +9-9. = 9 10/2 + 9 10/2. -9. = 81 Numerator/denominator 81(111111 111111)/81= 111111 111111 = 12345654321

This is a pretty simple problem, especially using a calculator.

The numerator is 999998000001 after multiplying.

The denominator, 1+2+3+4+5+6+7+8+9+8+7+6+5+4+3+2+1, is equal to 81.

Therefore, 999998000001/81 = 12345654321

Or am I not supposed to do this with a calculator???