Scientific Notation (write in scientific notation)

When dealing with very big or very small numbers, it's often convenient to compress them and make them uniform.  We use scientific notation to do this!

When writing in scientific notation, you move the decimal of any number to just after the first (highest) non-zero digit.  You then multiply that number by a multiple of 10 to expand it back to standard form. For more on the details of how scientific notation works, see Scientific Notation (Write in Standard Form).

What does that look like?

The radius of Earth is 150,000,000,000 meters.

That number is long, hard to read, and it would be easy to lose a zero.  

To write it in scientific notation, we follow the following process:

$$\eqalign{&150,000,000,000&&\text{Original number in standard form}\\&150,000,000,000.&&\text{Remember, when a number does not have a decimal, it's at the very end.}\\&1.\underleftarrow{50,000,000,000.}&&\text{Move the decimal so that there is only one digit to the left of it.}\\&1.\underset{\text{11 places to left}}{\operatorname{\underbrace{50,000,000,000.}}}&&\text{Count the number of places the decimal moved: this will become your exponent}\\&1.5 \times 10^{11}&&\text{Drop the zeros and multiply your new decimal times 10 to the power of however many spaces you moved the decimal}\\& && \text{Note: Your real number is bigger than your new decimal, so your exponent is positive}}$$

The process is the same for converting really tiny numbers to scientific notation (except, because your real number is smaller than the decimal in your scientific notation, you'll use a negative exponent).

The average red blood cell is .00072 centimeters in diameter.

That number is long, hard to read, and it would be easy to lose a zero.  

To write it in scientific notation, we follow the following process:

$$\eqalign{&.00072&&\text{Original number in standard form}\\&\underrightarrow{.0007.}2&&\text{Move the decimal so that there is only one non-zero digit to the left of it.}\\&\underset{\text{4 places}}{\operatorname{\underbrace{.0007}}}.2&&\text{Count the number of places the decimal moved: this will become your exponent}\\&7.2 \times 10^{-4}&&\text{Drop the zeros and multiply your new decimal times 10 to the power of however many spaces you moved the decimal}\\& && \text{Note: Your real number is smaller than your new decimal, so your exponent is negative}}$$

Overall, when you are given a  number and want to write it in scientific notation (in the real world, you do this for clarity and consistency), follow the following steps:

1. Locate the decimal (if you are working with a whole number and the decimal isn't written, write it in after the last digit of the number).
2. Move the decimal so that only one digit is to the left of it.
3. Count the number of places you moved the decimal.
4. Drop any extra zeros on the left or right of your number.
5. Write the number as the new decimal (with only one digit on the left of the decimal) times 10 raised to the power of the number of places you moved the decimal.
6. If your original number is smaller than your new decimal, your exponent will be negative.  If your original number is bigger than your new decimal, your exponent will be positive.

$\large{18,450,000 \Rightarrow 1.845 \times 10^7}$

and

$\large{.000098 \Rightarrow 9.8 \times 10^{-5}}$