How do you simplify #(4.65times10^-2)(5times10^6)#?

This is just one of several tricks. To multiply a decimal by 10 is strait forward. So lets see if we can incorporate that.

Not that 5 is the same as #10xx 1/2#

#color(blue)("Just dealing with the numbers")#

#4.65xx5 color(white)("ddd")->color(white)("ddd")4.65xx10xx1/2#

#color(white)("ddddddddd")->color(white)("ddd")46.5xx1/2#

#color(white)("ddddddddd")->color(white)("ddd") 23.25#

#color(white)("ddddddddd")->color(white)("ddd"color(green)(2.325xx10)#

There should only be one digit to the left of the decimal point and it is not 'allowed' to be 0.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Just dealing with the 10's part of "(4.65xx10^(-2))(5xxxx10^6))#

Believe it or not all we have to do is add the indeces

#10^(-2)xx10^6color(white)("d")=color(white)("d")10^(6-2)color(white)("d")=color(white)("d")color(green)(10^4)#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Putting it all together")#

#color(green)(2.325xx10xx10^4)#

This may be written as:

#2.325xx10^1xx10^4#

#2.325xx10^(1+4)#

#2.325xx10^5#