I'll multiply these two large numbers exactly using the standard multiplication algorithm.

123456789123456789 × 987654321987654321

Let me break this down systematically:

First, I'll recognize that these numbers have a pattern:
- 123456789123456789 = 123456789 × 10^9 + 123456789
- 987654321987654321 = 987654321 × 10^9 + 987654321

So I can write:
123456789123456789 = 123456789 × (10^9 + 1)
987654321987654321 = 987654321 × (10^9 + 1)

Therefore:
123456789123456789 × 987654321987654321 
= 123456789 × 987654321 × (10^9 + 1)^2
= 123456789 × 987654321 × (10^18 + 2×10^9 + 1)

First, let me calculate 123456789 × 987654321:
123456789 × 987654321 = 121932631112635269

Now I need:
121932631112635269 × (10^18 + 2×10^9 + 1)
= 121932631112635269 × 10^18 + 121932631112635269 × 2 × 10^9 + 121932631112635269

= 121932631112635269000000000000000000
+ 243865262225270538000000000
+ 121932631112635269

= 121932631356500531337905807112635269

Therefore:
**123456789123456789 × 987654321987654321 = 121932631356500531337905807112635269**
