## Chem – Multiplication and Division of Scientific Notation

What sections should I know before attempting to learn this section?

How do you multiply or divide scientific notation?

A lot of students run into trouble when they are trying to multiply or divide scientific notation. Dividing usually creates more problems than multiplying, but I will demonstrate both here just to be sure. Most of the students’ confusion in this area comes from a lack of understanding between their calculator and themselves. Different calculators have what you call different logic. That means different calculators read and interpret what you put into them in different ways. One of the most important lessons to learn early in class is to be comfortable and confident with your calculator. The best way to get good at that is keep trying to punch in the numbers you have in different orders until you get the correct answer.

Multiplying scientific notation is pretty straightforward. Whatever order the number appears in the problem you should punch them in the same order in your calculator. If you want to estimate your answer before you put it in the calculator to help guide you in case you make any mistakes, then you should add together exponents on the 10s when you are multiplying.

Dividing scientific notation can be more complicated.  When I meet my students for the first time about 95% of them have problems with dividing scientific notation, so it is nothing to be ashamed of and you are not the only one who is having trouble with it. Like I said before, most of the confusion with this part lies in the use of the calculator. The first and most importance piece of advice I can give a student when dividing scientific notation is to separate each step when doing the divisions. This may seem like a very slow way to go about it at first, but it is absolutely critical to your understanding of how this process works. It is also generally a good way to learn math throughout chemistry. Check out the demonstrated examples below.

In addition to having the abilities to solve multiplication and division problems separately, you also want to know how to solve them together.  The last demonstrated example shows you how to combine multiplication and division together.

Examples: Multiply the scientific notation below. VIDEO demonstration of the multiplication of the scientific notation below.

 (8.4 * 103) * (2.7 * 104) = 2.268 * 108 (1.5 * 10-2) * (7.6 * 105) = 1.14 * 104 (9.3 * 10-6) * (4.5 * 10-3) = 4.185 * 10-8

VIDEO Dividing Scientific Notation Demonstrated Example 1: Solve the division below.

 3.2 * 103 = 4.6 * 10-4

First step is to force ourselves to think differently about this problem. To do that we separate the decimal numbers from the 10 to the exponent numbers. This is still mathematically correct.

 3.2 103 = 4.6 10-4

Then we can divide each separately. Divide 3.2 by 4.6

 0.6956 103 = 10-4

Now clear your calculator. Then divide 103 by 10-4

 0.6956 107 = 1

Then multiply them back together

 0.6956 107 = 6.956 * 106 1

VIDEO Dividing Scientific Notation Demonstrated Example 2: Solve the division below.

 7.8 * 10-5 = 5.4 * 10-2

First step is to force ourselves to think differently about this problem. To do that we separate the decimal numbers from the 10 to the exponent numbers. This is still mathematically correct.

 7.8 10-5 = 5.4 10-2

Then we can divide each separately. Divide 7.8 by 5.4

 1.44 10-5 = 10-2

Now clear your calculator. Then divide 103 by 10-4

 1.44 10-3 = 1

Then multiply them back together

 1.44 10-3 = 1.44 * 10-3 1

VIDEO Multiplying and Dividing Scientific Notation Demonstrated Example 1: Solve the multiplication and division of scientific notation problems below.

 (6.9 * 103) * (8.3 * 10-2) = (8.7 * 10-4) * (1.4 * 107)

Separate all the numbers.

 6.9 103 8.3 10-2 = 8.7 10-4 1.4 107

Divide each section separately

 0.793 107 5.93 10-9 = 1

Then multiply them all together.

 0.793 107 5.93 10-9 = 4.7 * 10-2 1

PRACTICE PROBLEMS: Solve the multiplication and division of scientific notation problems below.

 (9.2 * 102) * (3.1 * 103) = 2.852 * 106 (5.4 * 10-3) * (2.6 * 107) = 1.404 * 105 (7.9 * 10-5) * (1.2 * 10-10) = 9.48 * 10-13 (3.4 * 10-6) * (9.6 * 10-12) = 3.264 * 10-17

 2.3 * 102 = 3.24 * 10-3 7.1 * 104

 9.6 * 10-3 = 4.0 * 102 2.4 * 10-5

 8.5 * 105 = 2.74 * 107 3.1 * 10-2

 1.8 * 10-7 = 4.62 * 10-14 3.9 * 106

 (9.5 * 102) * (4.1 * 103) = 6.36 * 10-4 (3.6 * 104) * (1.7 * 105)

 (6.7 * 10-4) * (8.3 * 10-2) = 3.69 * 104 (5.8 * 10-3) * (2.6 * 10-7)

 (7.3 * 1013) * (2.5 * 1015) = 3.56 * 1058 (3.2 * 10-18) * (1.6 * 10-12)

 (4.9 * 1014) * (2.7 * 10-20) = 6.96 * 10-3 (3.8 * 10-13) * (5 * 109)

Of course there are several different ways to think about multiplying and dividing scientific notation. However, the way I explained is the easiest for me to demonstrate and seems to be the easiest for students to learn. If your teachers show you a different way you can use that or you can use mine. In the end they are all the same in terms of the math.  