How do you write #350000# in scientific notation?

1 Answer
Aug 31, 2017

Answer:

#3.5 * 10^5#

Explanation:

Start by writing your number in standard notation

three hundred fifty thousand #-> 350,000#

Now, a number written in scientific notation will take the form

#color(white)(aa)color(blue)(m) xx 10^(color(purple)(n) color(white)(a)stackrel(color(white)(aaaaaa))(larr))color(white)(acolor(black)("the")acolor(purple)("exponent")aa)#
#color(white)(a/acolor(black)(uarr)aaaa)#
#color(white)(color(black)("the")acolor(blue)("mantissa")a)#

For normalized scientific notation, which is what you'll be dealing with in the vast majority of cases, you need to have

#1 <= |color(blue)(m)| < 10#

In your case, you start with

#350,000 * 10^0#

so you can say that you have

#color(blue)(m) = 350,000" "and " " color(purple)(n) = 0#

In order to write the number in scientific notation, you must divide it #10# as many times as you need in order to get

#1 <= color(blue)(m) < 10#

For every time you divide the number by #10#, you must also multiply it by #10# in order to keep its value unchanged.

The trick here is that you divide the mantissa by #10# and you multiply by #10# by increasing the exponent by #1#.

So, divide the mantissa by #10# and multiply

#(350,000)/10 * 10^0 * 10 = 35,000 * 10^1#

Since

#1 <= 35,000 color(red)(cancel(color(black)(<))) 10#

you must repeat the procedure.

#(35,000)/10 * 10^1 * 10 = 3,500 * 10^2#

Once again, you have

#1 <= 3,500 color(red)(cancel(color(black)(<))) 10#

so you must repeat the procedure again

#(3,500)/10 * 10^2 * 10 = 350 * 10^3#

Repeat it again

#350/10 * 10^3 * 10 = 35 * 10^4#

Repeat it again

#35/10 * 10^4 * 10 = 3.5 * 10^5#

This time, you have

#1 <= 3.5 < 10" "color(green)(sqrt())#

so you can say that your original number written in scientific notation will look like this

#350,000 = 3.5 * 10^5#

Notice that the mantissa keeps the same number of sig figs as the number written in standard form, i.e. #2# sig figs.