How do you find the scientific notation for #919,100#?

Assumption: The given value is #919100#

Our objective is to end up with a single, non zero integer to the left of a decimal point and everything else to the right of it.

In doing this we change the value. Consequently we have to show the mathematical adjustment needed to return it back to the original value.

Our target is to have the pre-adjusted value of;#" "9.191#

#color(brown)("~~~~~~~ Investigating the method ~~~~~~~~~~~~~~")#
#919100.0 xx 1/10 -> 91910.00#
#919100.0 xx 1/100 -> 9191.000#
#919100.0 xx 1/1000 -> 919.1000#
#919100.0 xx 1/10000 -> 91.91000#
#919100.0 xx 1/100000 -> 9.191000" "# This is the format we need

However in doin this we have changed its value. So how do we deal with this?

If you multiply any value by 1 you do not change its value. However, the value of 1 can take many forms. For example
#1=3/3=4/4 = (-9)/(-9)#. It can also take the form of:

#1=100000 /100000 # This can be our conversion factor.

#color(brown)("~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~")#
#color(blue)("Solving the question")#

Multiply 919100 by 1 but in the form of #1=100000/100000# giving

# 919100xx100000/100000" " =" " 919100/100000 xx100000#

Divide the denominator into the numerator giving

#9.19100 xx100000#

This can be written as

#color(blue)(" "9.19100 xx10^5)#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

#color(brown)("We have changed the way it looks without changing its intrinsic value")#

So #9.19100 xx10^5" is equivalent to "919100#

For being equivalent we write:

#9.19100 xx10^5-=919100#