8.81×106 times 3.47×103

Each of those is rounded to three significant digits.

(8.81)(3.47)×106+3

30.5707×109

So we round 30.5707 to three significant digits also:

30.6×109  

But that's not scientific notation for the first
number must be less than 10.

Write 30.6 as 3.06×101 

3.06×101×109

3.06×109+1 

3.06×1010    <--final answer

Let's analyze what happens here:

Both those numbers to multiply were rounded off
themselves, so:

The 8.81 could have been as large as 8.8149 and
the 3.47 could have been as large as 3.4749 and
the product as large as 30.63089601.

OR

The 8.81 could have been as small as 8.805 and
the 3.47 could have been as small as 3.465 and
the product as small as 30.509325.

They probably weren't both that large or both that small.
So let's average them:

The average of those two answers is 30.57011051, or 30.6.
so the only digits we can count as significant are the 3, 
the 0, and the 6.  The other digits to the right of those 
would be meaningless, so we drop them.
 
Edwin