The trick for scientific notations is that you have to place the decimal point after the first number you read.
#color(white)(aaaaaaaaaaaaa)#The way I remember it is like this:
#color(white)(aaaaaa)barul[|"I "color(blue)"left it bigger", "but you're "color(red)("right, it's smaller")|]#
Note: This works for both writing a number in scientific notation and when expanding it.
#--------------------#
#ul"Writing a number in scientific notation"#
When you need to write a number in scientific notation, you are going to have to take the decimal point and move it towards the right, counting how many times you move it until the decimal point goes after the first number you read.
#color(white)(aaaaaa)#
Taking our example,
#color(white)(aaaaa)0.0000000458stackrel"moving decimal place after first number"rarrcolor(orange)[4.58*10^-8#
you moved the decimal point 8 times to the #color(red)"right"# to place it after the #4#, your first number. And since you moved to the #color(red)("right")#, the exponent will be a #color(red)("smaller number")# than #0#, meaning a negative number.
#color(white)(aaaaaaaaaaaaaaa)color(orange)barul[| 4.58*10^ -8|] " Moving to the right gave us"#
#color(white)(aaaaaaaaaaaaaaaaaaaaaaaaaaa)"a negative exponent"#
#--------------------#
#ul"Expanding out a number in scientific notation"#
When you need to expand the scientific notation, you are going to have to take wherever the decimal point is and move it either to the left or to the right depending on what the exponent is. From there, we take whatever our exponent is and get it back to #0#.
#color(white)(aaaaaa)#
So taking our previous answer as an example,
#color(white)(aaaaaaaaaaaaaaaaaaa)4.58*10^-8#
we see that our exponent is #"-8"#. We want to take this exponent and get it back to #0#. Going from #"-8" to 0# means we are getting #color(blue)["bigger"# in value. If we are getting #color(blue)["bigger"#, then from our mnemonic,
#color(white)(aaaaaa)barul[|"I "color(blue)"left it bigger", "but you're "color(red)("right, it's smaller")|]#
we must move to the #color(blue)("left")# of #4.58#.
#ul"Expanding"#
#4color(magenta)"."58color(white)(aaaaaaaaaaa)(-8)darr#
#color(magenta)"."458color(white)(aaaaaaaaaaa)(-7)darr#
#color(magenta)".0"458color(white)(aaaaaaaaaa)(-6)darr#
#color(magenta)".00"458color(white)(aaaaaaaaa)(-5)darr#
#color(magenta)".000"458color(white)(aaaaaaaa)(-4)darr#
#color(magenta)".0000"458color(white)(aaaaaaa)(-3)darr#
#color(magenta)".00000"458color(white)(aaaaaa)(-2)darr#
#color(magenta)".000000"458color(white)(aaaaa)(-1)darr#
#color(orange)[.0000000458]color(white)(aaaaaa)(0)" complete"#
#color(white)(aaaaaaaaa)4.58*10^-8stackrel"after expanding"rarrcolor(orange)barul[| 0.0000000458 |#
