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## 8th grade

### Course: 8th grade > Unit 1

Lesson 11: Arithmetic with numbers in scientific notation- Multiplying & dividing in scientific notation
- Multiplying three numbers in scientific notation
- Multiplying & dividing in scientific notation
- Subtracting in scientific notation
- Adding & subtracting in scientific notation
- Simplifying in scientific notation challenge

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# Multiplying & dividing in scientific notation

In order to simplify multiplication and division using scientific notation, you should multiply and divide numbers with the same base, and add or subtract the exponents. Through this process, complex expressions can be simplified into a single value multiplied with 10 to a certain power. As an example, 7 times 10 to the fifth over 2 times 10 to the negative 2 times 2.5 times 10 to the ninth can be simplified to 1.4 times 10 to the negative 2. Scientific notation helps to simplify complex equations that involve multiplying and dividing numbers with the same base. Created by Sal Khan.

## Want to join the conversation?

- I need help how did you get 5 from 2 x 2.5 on my calculator i got 4.5(0 votes)
- my friend, you put a + instead of a x... 2 + 2.5 = 4.5 --- 2 x 2.5 = 5 .(34 votes)

- Does the first factor in the answer need to have a decimal point in it always? Or are there exceptions for problems where if you simplify them another way, the first digit ends up being zero? My question isn't specific to this video, just a general question on this topic.(4 votes)
- The format for scientific notation is that there will always be just 1 digit to the left of the decimal point and that digit can not be zero.

For example:

9,300,000 becomes 9.3 x 10^6

0.0005 becomes 5 x 10^(-4)

Hope this helps.(18 votes)

- Is there a reason why one puts the point right after the first significant figure when using scientific notation? E.g. 245324.321 as 2.45324321*10^5 instead of 245.324321*10^3? I think the last notation is smarter because it is easier to see that the number is a thousand-something.(5 votes)
- In scientific notation, the number has to be from 1 - 9. For example, 3.1415926 x 10^7 is correct instead of 31.415926 x 10^6 which is incorrect.

I know this is a late response, but I hope this helped anyone!(5 votes)

- I get the multiplication, but the division looks like common core. 😳😔😫😡(6 votes)
- i hate math bruh(2 votes)

- What do I do if the product/quotient is not appropriate for scientific notation? For example (5.0 x 10^1) x (2.0 x 10^1) which I imagine equals 10 x 10^2. If I had to guess I'd say increase 10^2 to 10^3 and make 10 to 1.0 so that it'd be 1.0 x 10^3. Sorry for any poor wording(4 votes)
- Your thinking process is correct as is your answer(5 votes)

- Did Sal make an error here? At2:31, he is checking to see if 1.4 * 10^-2 is expressed in scientific notation. After confirming that 1.4 is greater than or equal to one, he next asks if it is less than or equal to nine.

But the rule for scientific notation is that the decimal portion of the number must be less than (and not equal to ) 10. If I wrote a number like 9.9 * 10^-2, this would be a decimal that is*not*less than or equal to nine, but it would be in scientific notation, because the rule is that the decimal must be greater than or equal to one and less than 10.(5 votes)- Yes you are correct and you've made a good point. Great catch!(3 votes)

- At1:15, Sal says that his fraction can be viewed as 7/5 * 10^5/10^7. Is there any section or video where I can learn the things that fractions can do and how to work well with them?(3 votes)
- Search up 'Fractions' and choose the section you want to learn: for example, 'Multiplying and Dividing Fractions'.(5 votes)

- I hate this I put the answer as 4 in one of the questions and it said it was wrong. It said the correct answer was something else that was equal to 4.(4 votes)
- If you post the actual problem, them someone could help you. Without the details its hard to say what was wrong.

Did you look at the hints? I find them useful if something is marked wrong and I think I have the correct value. If you really think the website has an error, you need to use the "report a problem" link within the exercise set.(3 votes)

- i get the multiply but the division is hella confusing💵💵(5 votes)
- Can you subtract exponents if the base is not the same?(2 votes)
- When dividing expressions that use exponents, we can subtract exponents only when the bases are the same.(3 votes)

## Video transcript

We have 7 times 10 to
the fifth over 2 times 10 to the negative 2 times
2.5 times 10 to the ninth. So let's try to simplify
this a little bit. And I'll start off by trying to
simplify this denominator here. So the numerator's just
7 times 10 to the fifth. And the denominator,
I just have a bunch of numbers that are being
multiplied times each other. So I can do it in any order. So let me swap the order. So I'm going to do over
2 times 2.5 times 10 to the negative 2
times 10 to the ninth. And this is going to be equal
to-- so the numerator I haven't changed yet-- 7 times
10 to the fifth over-- and here in the
denominator, 2 times-- let me do this in
a new color now. 2 times 2.5 is 5. And then 10 to the negative
2 times 10 to the ninth, when you multiply
two numbers that are being raised to exponents
and have the exact same base-- so it's 10 to the negative 2
times 10 to the negative 9-- we can add the exponents. So this is going to be
10 to the 9 minus 2, or 10 to the seventh. So times 10 to the seventh. And now we can
view this as being equal to 7 over 5 times
10 to the fifth over 10 to the seventh. Let me do that in
that orange color to keep track of the colors. 10 to the seventh. Now, what is 7 divided by 5? 7 divided by 5 is equal to--
let's see, it's 1 and 2/5, or 1.4. So I'll just write it as 1.4. And then 10 to the fifth
divided by 10 to the seventh. So that's going to be
the same thing as-- and there's two
ways to view this. You could view this as
10 to the fifth times 10 to the negative 7. You add the exponents. You get 10 to the negative 2. Or you say, hey, look,
I'm dividing this by this. We have the same base. We can subtract exponents. So it's going to be
10 to the 5 minus 7, which is 10 to the negative 2. So this part right over here is
going to simplify to times 10 to the negative 2. Now, are we done? Have we written what we have
here in scientific notation? It looks like we have. This value right over here is
greater than or equal to 1, but it is less
than or equal to 9. It's a digit between 1
and 9, including 1 and 9. And it's being multiplied
by 10 to some power. So it looks like we're done. This simplified to 1.4
times 10 to the negative 2.