# Scientific Notation Multiplication and Division

When it comes to these two operations, students often get intimidated by before even seeing a single demonstration of how to solve these types of problems. Once they do, they figure out it is a cinch. You treat the decimal part of the operation as would normally. The magic comes in when you concerned with the base ten section. When multiplying, you simply add the powers that are present. Division, being the counter operation to multiplication, you do the opposite and subtract them. That is all there is to it. It really just comes down to understanding the operation that is taking place. These worksheets show students how to find the outcome of quotients and products of values that are expressed in scientific notation.

### Aligned Standard: 8.EE.A.4

- Multiplying Scientific Notation Step-by-Step Lesson- I love how you can just add the powers of ten. The first person to figure that out must have been floored.
- Guided Lesson - We add two quotients in here to get more practice in. Remember to pay attention to the exponents.
- Guided Lesson Explanation - Finding quotients is very similar to finding products when working with this form of notation.
- Practice Worksheet - When we multiply forms of scientific notation you get some interesting products in here.
- Matching Worksheet - The answers to these are not as obvious as you would think.

- Answer Keys - These are for all the unlocked materials above.

### Homework Sheets

The top half of all the worksheets is filled with multiplication. The bottom portions of the sheets are where you will be processing division problems.

- Homework 1 - Separate the problem into its two components and then break the pieces into parts.
- Homework 2 - Writing the number in proper scientific form. You will practice with both operations in here.
- Homework 3 - Find the quotient of the powers of ten by subtracting. This is tricky at first.

### Practice Worksheets

A good idea is to get in the habit of circling and rewriting the exponents immediately.

- Practice 1 - Separate the problem into its two components. a) Constants b) Powers of ten
- Practice 2 - A sample problem that you will be working with here: (1.17 × 10
^{3}) (2.11 × 10^{3}) - Practice 3 - The important part is following the sections of the notation before you start to multiply or divide it.

### Math Skill Quizzes

I used a larger font for the numbers that forces kids to rewrite the problem. This is a good thing!

- Quiz 1 - Rearrange the problem to put the constants and powers of ten together.
- Quiz 2 - We follow the same exact strategy we need to. See how you are doing with this skill.
- Quiz 3 - Complete all the problems to get them all headed in the right direction.

### How to Multiply and Divide Values in Scientific Notation

Multiplying and dividing values with scientific notations is easier since the values have the same base 10 in them. Let us solve both of them separately with the help of examples.

**Multiplication Example** - (5.6 × 10^{5}) × (2.4 × 10^{5}).

Here we will first have to separate the decimal number and the base 10 number and multiply them.

(5.6 × 2.4) × (10^{5} × 10^{5})

The exponential numbers have the same base (i.e., 10), so according to the multiplicative property, the powers are added. 10^{5} × 10^{5} = 10^{10}. Now the decimal numbers will have a result of 13.44 (5.6 × 2.4).

The resulting value would be: (13.44 × 10^{10}).

**Division Example** - ((5.6 × 10^{5})/(4.3 × 10^{4})).

Here, similar to the multiplication operations, we can use an exponential property of division to divide the notation by subtracting the powers that are present: ((5.6 × 10 ^{(5-4)}) / (4.3 ))

Now, you can divide the decimal numbers (5.6 / 4.3) to get an answer of 1.3, and the answer becomes:

1.3 × 10^{1} = 13

### When Would You Find Yourself Multiplying and Dividing Scientific Notation?

As the name implies, scientific notation is form of display numeric values that the science community gravitates towards. This is because they are constantly studying and researching things that are either very large or very small. Chemist will often use this number expression form to indicate the amount of a substance that they use to create a reaction. When they need to calculate how much of the substance is needed to be present in order for the reaction to take place, they will need to either multiply or divide those values that are stated in scientific notation. There is an example of using operations with scientific notation when the values are ridiculously small, and another example would be Microbiologists that study the effects of a drug on cells. They will often place the remnants of the drug on a plate with bacterial cells to see how they react to the drug. After some time, they will come back and evaluate the number of bacteria that survived the drug treatment. To determine the mortality rate of the drug they will often divide the surviving population by the starting population in scientific notation. Astronomers work with numbers that are in the opposite spectrum, they are really big numbers. They will often use these distances to measure the size of stars or solar systems. To determine the relative distances of solar systems between one another they will often multiply and divide their distances in scientific notation.