Let's dive quickly into the operations of arithmetic using scientific notation.

## How to add?

The addition in scientific notation can be done by following very simple rules:

- Change all numbers to the same power of 10.
- Add the coefficients and put the common power of 10 as \(\times 10^n\).
- Convert to scientific notation again if there is not only one nonzero number to the left of decimal point. If you need to do this, change or add the exponents again (apply exponents rule).

### Example

You have two numbers \(2.4 \times 10^3\) and \(5.71 \times 10^5\). To add these two numbers easily, you need to change all numbers to the common power of 10. You can change exponent of any number. Here we change the exponent in \(5.71 \times 10^5\) to 3 and it is \(571 \times 10^3\) (note the decimal point moved two places to the right).

\[\begin{align*} 2.4 \times 10^3 + 5.71 \times 10^5 \\ 2.4 \times 10^3 + 571 \times 10^3 \\ (2.4 + 571) \times 10^3 \\ 573.4 \times 10^3 \\ 5.734 \times 10^2 \times 10^3\\ 5.734 \times 10^{2+3} \\ 5.734 \times 10^5 \end{align*}\]

Now simply add coefficients, that is 2.4 + 571 and put the power 10, so the number after addition is \(573.4 \times 10^3\). The final step is to convert this number to the scientific notation. Note that the coefficient must be greater than 1 and smaller than 10 in scientific notation. So the number in scientific notation after the addition is \(5.734 \times 10^5\).

You do not need to convert the final number into scientific notation again if you have changed exponent in \(2.4 \times 10^3\) to 5, so it is a good idea to convert smaller exponent to greater exponent. After subtracting the two exponents 5 - 3 you get 2 and the 2 to the power of 10 is 100. So 2.4 needs to be divided by 100 or the decimal point needs to be moved two places to the left, and that gives 0.024. Now we have the same exponent in both numbers. Just add 0.024 + 5.71 which gives 5.734 and the result is \(5.734 \times 10^5\).

\[\begin{align*} 2.4 \times 10^3 + 5.71 \times 10^5 \\ 0.024 \times 10^3 + 5.71 \times 10^5 \\ (0.024 + 5.71) \times 10^5 \\ 5.734 \times 10^5 \\ \end{align*}\]

## How to multiply?

Multiplication of numbers in scientific notation is easy. Simply multiply the coefficients and add the exponents. If necessary, change the coefficient to number greater than 1 and smaller than 10 again.

### Example

Here we have two numbers \(7.23 \times 10^{34}\) and \(1.31 \times 10^{11}\). When you multiply these two numbers, you multiply the coefficients, that is \(7.23 \times 1.31 = 9.4713\). Then all exponents are added, so the exponent on the result of multiplication is \(11+34 = 45\). The final result after the multiplication is \(9.4713 \times 10^{45}\) or the process is shown below:

\[\begin{align*} (7.23 \times 10^{34}) \times (1.31 \times 10^{11}) \\ = 7.23 \times 1.31 \times 10^{34} \times 10^{11} \\ = 9.4713 \times 10^{34 + 11}\\ = 9.4713 \times 10^{45} \end{align*}\]

Apply the exponents rule and voila! If the coefficient in the result is greater than 10 convert that number to greater than 1 and smaller than 10 by changing the decimal location and add the exponents again.

## How to divide?

The division of two scientific numbers is similar to multiplication but in this case we divide coefficients and subtract the exponents.

You have two numbers \(1.03075 \times 10^{17}\) and \(2.5 \times 10^5\) . To divide these numbers we divide 1.03075 by 2.5 first, that is 1.03075/2.5 = 0.4123. Then we subtract the exponents of these numbers, that is 17 - 5 = 12 and the exponent on the result of division is 12.

Note that the number 0.4123 is less than 1, so we make this number greater than 1 and smaller than 10. To do that the decimal point goes between 4 and 1 and the number of steps we moved to the right across the digits to our new location is subtracted from the exponent of 10. So, The final exponent of 10 is \(12 - 1 = 11\) and the number is 4.123. So the result is \(4.123 \times 10^{11}\). Or mathematically,

\[\begin{align*} \frac{1.03075 \times 10^{17}}{2.5 \times 10^5} &= \frac{1.03075}{2.5} \times 10^{17 - 5} \\ &= 0.4123 \times 10^{12} = 4.123 \times 10^{-1} \times 10^{12} \\ &= 4.123 \times 10^{-1+12} = 4.123 \times 10^{11} \end{align*}\]