I need to multiply these two large numbers exactly.

Let me break this down systematically:

123456789123456789 × 987654321987654321

I notice both numbers have a repeating pattern:
- 123456789123456789 = 123456789 × 10^9 + 123456789
- 987654321987654321 = 987654321 × 10^9 + 987654321

So if I let:
- a = 123456789
- b = 987654321

Then I'm computing:
(a × 10^9 + a) × (b × 10^9 + b) = (a(10^9 + 1)) × (b(10^9 + 1))
= ab(10^9 + 1)^2

First, let me find ab:
123456789 × 987654321 = 121932631112635269

Next, (10^9 + 1)^2:
(10^9 + 1)^2 = 10^18 + 2×10^9 + 1 = 1000000002000000001

Now multiply:
121932631112635269 × 1000000002000000001

= 121932631112635269 × 10^18 + 121932631112635269 × 2 × 10^9 + 121932631112635269

= 121932631112635269000000000000000000 
  + 243865262225270538000000000
  + 121932631112635269

= **121932631112635269243865262225270538121932631112635269**
