Since it's important that you know how to perform scientific notation math without requiring a calculator -- especially if you may be required to show your work -- I have included a short lesson for each operation on this page. Note that each lesson is reflective of the results that will appear in the calculated results.
Multiplying Scientific Notation
To multiply two SN's you basically just multiply the coefficients and add the exponents, and then convert that result to proper scientific notation format (if not already).
Example multiplication problem:
(3 x 106) x (6 x 104)
Here are the steps to solving the above problem.
| Multiplication Steps |
| (3 x 106) x (6 x 104) |
| Group like terms: |
| = (3 x 6) x (106 x 104) |
| Multiply coefficients and add exponents: |
| = (18) x (1010) |
| Convert to proper SN: |
| = 1.8 x 1011 |
The other way to multiply scientific notations is to convert the notations to real numbers, perform the multiplication, and then convert the number back to scientific notation, like the following:
| Alternative Multiplication Method |
| (3 x 106) x (6 x 104) |
| = (3 x 1000000) x (6 x 10000) |
| = 3000000 x 60000 |
| = 180000000000 |
| = 1.8 x 1011 |
Of course, this alternative method can be cumbersome if the numbers you are working with are very large or very small, and may not be possible if the calculator you are using won't display the results in regular notation.
Dividing Scientific Notation
To divide two SN's you divide the coefficients and subtract the trailing exponent from the leading exponent, and then convert that result to proper scientific notation format (if not already).
Example division problem:
(5 x 108) ÷ (2 x 104)
Here are the steps to solving the above problem.
| Division Steps |
| (5 x 108) ˜ (2 x 104) |
| Group like terms: |
| = (5 ÷ 2) x (108 ÷ 104) |
| Divide coefficients and Subtract exponents: |
| = (2.5) x (104) |
| Convert to proper SN: |
| = 2.5 x 104 |
The other way to divide scientific notations is to convert the notations to real numbers, perform the division, and then convert the number back to scientific notation, like the following:
| Alternative Division Method |
| (5 x 108) ÷ (2 x 104) |
| = (5 x 100000000) ÷ (2 x 10000) |
| = 500000000 ÷ 20000 |
| = 25000 |
| = 2.5 x 104 |
Adding Scientific Notation
To add two SN's you factor out one of the powers of 10, convert the remaining scientific notations to real numbers, add the real numbers, and then convert that result to proper scientific notation format (if not already).
Example addition problem:
(6 x 104) + (7 x 103)
Here are the steps I use to solve the above addition problem.
| Addition Steps |
| (6 x 104) + (7 x 103) |
| Factor out 1 of the powers of 10 |
| = (6 x 104/104 + 7 x 103/104) x 104 |
| Perform division of exponents: |
| = (6 x 100 + 7 x 10-1) x 104 |
| Convert SN's to real numbers: |
| = (6 + 0.7) x 104 |
| Combine real numbers: |
| = (6.7) x (104) |
| Convert to proper SN: |
| = 6.7 x 104 |
There is a second method for solving the above addition problem (not included in the calculated results), which requires converting the numbers to the same power of 10. To do that you move the decimal point of one coefficient to the left or right the number of places needed to make that number's power of 10 equal to the other. You then add the resulting coefficients while keeping the power of 10 the same.
| Addition Steps |
| (6 x 104) + (7 x 103) |
| Convert numbers to same exponents: |
| = (6 x 104) + (.7 x 104) |
| Add coefficients and keep the exponent the same: |
| = (6.7 x 104) |
| Convert to proper SN: |
| = 6.7 x 104 |
A third way to add scientific notations (if the numbers aren't too large or small) is to convert the notations to real numbers, perform the addition, and then convert the number back to scientific notation, like the following:
| Alternative Addition Method |
| (6 x 104) + (7 x 103) |
| = (6 x 10000) + (7 x 1000) |
| = 60000 + 7000 |
| = 67000 |
| = 6.7 x 104 |
Subtracting Scientific Notation
To subtract one SN from another, you first factor out one of the powers of 10, convert the remaining scientific notations to real numbers, subtract the second real number from the first, and then convert that result to proper scientific notation format (if not already).
Example subtraction problem:
(7 x 106) - (4 x 103)
Here are the steps I use to solve the above subtraction problem.
| Subtraction Steps |
| (7 x 106) - (4 x 103) |
| Factor out 1 of the powers of 10 |
| = (7 x 106/106 - 4 x 103/106) x 106 |
| Perform division of exponents: |
| = (7 x 100 - 4 x 10-3) x 106 |
| Convert SN's to real numbers: |
| = (7 - 0.004) x 106 |
| Perform real number subtraction: |
| = (6.996) x (106) |
| Convert to proper SN: |
| = 6.996 x 106 |
There is a second method for solving the above subtraction problem (not included in the calculated results), which requires converting the numbers to the same power of 10. To do that you move the decimal point of one coefficient to the left or right the number of places needed to make that number's power of 10 equal to the other. You then subtract the resulting coefficients while keeping the power of 10 the same.
| Addition Steps |
| (7 x 106) - (4 x 103) |
| Convert numbers to same exponents: |
| = (7 x 106) - (.004 x 106) |
| Subtract coefficients and keep the exponent the same: |
| = (6.996 x 106) |
| Convert to proper SN: |
| = 6.996 x 106 |
The third way to subtract one scientific notation from another (if numbers aren't too large or small) is to convert the notations to real numbers, perform the subtraction, and then convert the number back to scientific notation, like the following:
| Alternative Subtraction Method |
| (7 x 106) - (4 x 103) |
| = (7 x 1000000) - (4 x 1000) |
| = 7000000 - 4000 |
| = 6996000 |
| = 6.996 x 106 |
Double Check the Answer
Since the programming language I use to create online calculators is not well-equipped to handle very large or very small numbers, I had to come up with several bizarre workarounds to overcome the language's short-comings.
For this reason, the calculator provides two answers: one returned by my workarounds, and one returned by the programming language. If these two answers are different beyond just rounding issues, please use the feedback form beneath the calculator to let me know the notations you entered and the operation you selected so I can investigate the difference.
