# How do you write 350000 in scientific notation?

Aug 31, 2017

$3.5 \cdot {10}^{5}$

#### Explanation:

Start by writing your number in standard notation

three hundred fifty thousand $\to 350 , 000$

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

$\textcolor{w h i t e}{a a} \textcolor{b l u e}{m} \times {10}^{\textcolor{p u r p \le}{n} \textcolor{w h i t e}{a} \stackrel{\textcolor{w h i t e}{a a a a a a}}{\leftarrow}} \textcolor{w h i t e}{a \textcolor{b l a c k}{\text{the")acolor(purple)("exponent}} a a}$
$\textcolor{w h i t e}{\frac{a}{a} \textcolor{b l a c k}{\uparrow} a a a a}$
$\textcolor{w h i t e}{\textcolor{b l a c k}{\text{the")acolor(blue)("mantissa}} a}$

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

$1 \le | \textcolor{b l u e}{m} | < 10$

$350 , 000 \cdot {10}^{0}$

so you can say that you have

$\textcolor{b l u e}{m} = 350 , 000 \text{ "and " } \textcolor{p u r p \le}{n} = 0$

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

$1 \le \textcolor{b l u e}{m} < 10$

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

$\frac{350 , 000}{10} \cdot {10}^{0} \cdot 10 = 35 , 000 \cdot {10}^{1}$

Since

$1 \le 35 , 000 \textcolor{red}{\cancel{\textcolor{b l a c k}{<}}} 10$

you must repeat the procedure.

$\frac{35 , 000}{10} \cdot {10}^{1} \cdot 10 = 3 , 500 \cdot {10}^{2}$

Once again, you have

$1 \le 3 , 500 \textcolor{red}{\cancel{\textcolor{b l a c k}{<}}} 10$

so you must repeat the procedure again

$\frac{3 , 500}{10} \cdot {10}^{2} \cdot 10 = 350 \cdot {10}^{3}$

Repeat it again

$\frac{350}{10} \cdot {10}^{3} \cdot 10 = 35 \cdot {10}^{4}$

Repeat it again

$\frac{35}{10} \cdot {10}^{4} \cdot 10 = 3.5 \cdot {10}^{5}$

This time, you have

$1 \le 3.5 < 10 \text{ } \textcolor{g r e e n}{\sqrt{}}$

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

$350 , 000 = 3.5 \cdot {10}^{5}$

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