Question

How do you write `350000` in scientific notation?

Answer 1

`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.