To compute $987654321^2$ exactly, we multiply $987654321 \times 987654321$.
We can write $987654321 = 10^9 - 12345679$.
We can also rewrite it as $(10^9 - 12345679)$.
Then we have
$$987654321^2 = (10^9 - 12345679)(10^9 - 12345679)$$
$$= (10^9)^2 - 2(10^9)(12345679) + (12345679)^2$$
$$= 10^{18} - 24691358 \times 10^9 + (12345679)^2$$
Now, we can compute $(12345679)^2$ without using a calculator.

Alternatively, we can write $987654321 = 10^9 - 12345679$.
Then $987654321 = \frac{8888888888}{9}$.
Then $987654321^2 = (\frac{8888888888}{9})^2 = \frac{790123456790123456}{81}$
$987654321^2 = (\frac{8888888888}{9})^2$
$987654321^2 = (\frac{8888888888}{9}) \times (\frac{8888888888}{9})$
This method is messy.
Let $x = 987654321$.
Then $x^2 = (987654321)^2 = 975463078456$.
Consider the pattern:
$1^2 = 1$
$11^2 = 121$
$111^2 = 12321$
$1111^2 = 1234321$
$11111^2 = 123454321$
$111111^2 = 12345654321$
$1111111^2 = 1234567654321$
$11111111^2 = 123456787654321$
$111111111^2 = 12345679012345679012345678^2= 12345678987654321$
$9^2 = 81$
$99^2 = 9801$
$999^2 = 998001$
$9999^2 = 99980001$
$987654321^2 = 975308641975308641$.

We can write $x = 10^9 - 12345679$.
Then $x^2 = (10^9 - 12345679)^2 = 10^{18} - 2(12345679)(10^9) + (12345679)^2$
$10^{18} - 2469135800000000000 + 15241578750190521$
If we do this by hand, we get $975308641^2$.
$987654321^2 = 975308642006321$
$= 975308641975308641$.

$987654321^2 = 975308642006321$.
$987654321^2 = 975308641975308641$
$975308641975308641$

Final Answer: The final answer is $\boxed{975308641975308641}$
