# Adding & Subtracting in Scientific Notation

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• 0:00 What Is Scientific Notation?
• 1:04 Adding in Scientific Notation
• 4:32 Subtracting in…
• 6:40 Lesson Summary
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Lesson Transcript
Instructor: Julie Zundel

Julie has taught high school Zoology, Biology, Physical Science and Chem Tech. She has a Bachelor of Science in Biology and a Master of Education.

Scientific notation helps scientists write really big or small numbers with ease, but adding and subtracting using scientific notation requires a few steps. This lesson will outline the rules you need to add and subtract in scientific notation.

## What Is Scientific Notation?

The Earth is about 93,000,000 miles from the sun, give or take some miles, depending on the orbit. A tiny virus is about 0.000002 centimeters in diameter. As a scientist, you would find yourself consistently working with really big numbers, like 93 million, or really small numbers, like the diameter of a teeny, tiny virus. The chances of making a mistake are pretty high when you have to constantly write all of those zeros. Not to mention, adding and subtracting with all of those zeros is somewhat cumbersome.

So scientists use scientific notation to represent really big or really small numbers. Take a look at this image to see what scientific notation looks like.

The first number is the coefficient (sometimes it is referred to as a constant); the ten is called the base; and the tiny number above the base is the exponent.

Take a look at the table so you can familiarize yourself with how numbers are represented using scientific notation.

Decimal Number Scientific Notation
400 4x10^2
40 4x10^1
4 4x10^0
0.4 4x10^-1
0.04 4x10^-2

Now that you have an idea of what scientific notation is, let's solve some problems so you can learn to add in scientific notation. But first, let's review the rules:

• In order to add in scientific notation, the exponents must be the same.
• In the final answer, the coefficient must be between 1 and 10.

OK. Let's get started!

### Example 1

Let's look at the following problem: (1.2 x 10^3) + (2.4 x 10^3).

The exponents are the same, so we are good to go.

1. Start by adding the coefficients: 1.2 + 2.4 = 3.6
2. You leave the base and the exponents alone
3. The final answer is 3.6 x 10^3

As you can see, the final coefficient is between 1 and 10. If it weren't, we'd need to move the decimal and adjust the exponent (we'll see that in the next example).

### Example 2

Let's do a more complicated example: (9.2 x 10^4) + (5.3 x 10^4).

You can see the exponents are the same, so you can start by adding the coefficients like before.

1. 9.2 + 5.3 =14.5
2. Leave the base and exponents the same, so you end up with 14.5 x 10^4.
3. You can see the coefficient is not between 1 and 10, so we have an additional step
4. Move the decimal in 14.5 to the left by 1. This gives you 1.45. But, because you moved the decimal one to the left, you must increase the exponent by 1. Your answer is 1.45 x 10^5.

Remember, when you move the decimal, adjust the exponent. Moving the decimal to the left increases the exponent by one per place moved. Moving the decimal to the right decreases the exponent by one per place moved.

### Example 3

This time you have the problem: (4.5 x 10^4) + (2.1 x 10^6). Uh oh! You can see the exponents are not the same. We need to make them the same before we can proceed. A good rule is to make the smaller exponent the same as the larger exponent. So, to do this, you will need to:

1. Change 4.5 x 10^4 to the same number, but x 10^6. But if you change the exponent by 2, you need to change the decimal in the coefficient. Remember, if the exponent increases by two, the decimal gets moved to the left by two places.
2. Move the decimal to the left so you end up with 0.045 x 10^6

Now you can proceed with the problem like before.

1. Add the coefficients: 0.045 + 2.1 = 2.145
2. Leave the base and exponents the same
3. Your answer is: 2.145 x 10^6, and since 2.145 is between 1 and 10, you're all done!

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