In general, the order in which multiplication is done is irrelevant - the answer is always the same. So, we can rewrite this problem as (1.97 x 4) x (108 x 109).
The first part - (1.97 x 4) - equals 7.88.
The second part - (108 x 109) - equals 1017.
The end result would be 7.88 x 1017.
The second part may require a bit of explanation.
When a number such as 1000 is multiplied by 100, we essentially add the zeroes. The first number has three zeroes, the second has two zeroes. Their product is 100,000, which has five zeroes. This problem written in exponential form is 103 x 102 = 105. Essentially, we add the two exponents to each other to get the result (103+2 = 105).
So in the case of 108 x 109, it is just a question of adding all the zeroes together. The result is 108+9 or 1017.
To conceptualize this, think of our odometer. (See the essay under "About Big Numbers.") When we multiply these numbers we are simply adding drums to our odometer. 108 multiplied by 109 means adding nine new drums to the existing eight. The total number of drums on the new odometer is seventeen.