Hello!
As you have shown, scientific notation takes the form:
m * 10n, with m as a real number and n as an integer
Multiplying two numbers in scientific notation together looks like:
(m1*10n1)*(m2*10n2)
and because we are allowed to multiply numbers together in any order and get the same result, we can rearrange this to:
(m1*m2)*(10n1*10n2)
So we can say that multiplying two numbers in scientific notation is just the product of the real parts times the product of the 10n parts.
For the problem listed above, this would mean that the product of those two numbers is:
(5*8.4)*(10-11*108)
From the rules of algebra, we have that (am*an) = am+n
So the above product simplifies to:
(5*8.4)*(10(-11+8)) = 42.0*10-3 m
The 10n term indicates which way to move the decimal point and how many spaces to move it. A negative exponent means move the decimal point to the left n places, and a positive exponent means move it to the right n places. So 42.0*10-3 would equal 0.042 m, if we were to write it conventionally (0.042 is 42.0 with the decimal point moved three places to the left, as the -3 exponent specifies)
42.0*10-3 is often sufficient for how the result is written in physics. But there is sometimes a stricter definition of scientific notation, where a decimal point has to be between the two leftmost significant figures. So, in this convention, instead of writing
42.0*10-3 m
one would move the decimal point between the "4" and the "2," and adjust the exponent of the 10n term accordingly, to still get a result of 0.042. This would mean making the answer:
4.2*10-2 m
These mean exactly the same thing. It is up to your class and instructor which convention you work with.
Hope this helps! Let me know if you have any questions about this or any related topics.
Steven W.
07/27/16