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Lesson 11: Arithmetic with numbers in scientific notation

# Multiplying three numbers in scientific notation

Multiplying really big or really small numbers is much easier when using scientific notation. When we multiply numbers in scientific notation, we can commute and associate the factors. That lets us multiply the decimal factors first. Then we can use exponent properties to multiply the powers of 10. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

• Why do we need scientific notation in chemistry?
• As an example, think of Avogadro's number. It's a huge number, and it would be tedious to write it out in calculations and etc. Sal actually discusses it in his 'Intro to Scientific Notation' video, I believe.

I hope this helped!
• My eighth grade teacher is making us doing this...how do i get rid of the exponent? Or can i not do that?
• You can get rid of the exponent by converting the number and turning it into a decimal.
You do this by taking the exponent and listing that number of zeros behind the first part of the number

So the number 3 x 10^4 would be 30000

If the first part of the number has multiple digits then you subtract however many digits there are from the zeroes.
So 3.12 x 10^4 would be 31200
• why do we need to use scientific notation at all?
• Scientific notation is really useful when you are dealing with really big numbers (like the distance between stars and planets in astronomy) and really small numbers (like the mass of atoms and molecules in chemistry).

Using scientific notation allows you to do calculations without having to deal with all the pesky zeros.

Hope this helps!
• When do you know if you have to use scientific notation or just rounding"
e.g IN 119.5
Is it the same 1.19x10(2) as 120?
• You use scientific notation a lot in chemistry or physics along with significant figures (a type of rounding, sort of). Usually, in math, you only round as told.

Scientific notation is helpful when you need to use significant figures because it makes it easy to tell how many there are. For instance, 6.00x10^3 has 3, but 6000 has 1. If you haven't learned about significant figures, it's a way to measure how precise your answer can be given how precise the given numbers are. It's assumed that 6000 is rounded to the nearest thousand, but the 0s in 6.00 mean it's accurate to that position, so 6.00x10^3 is rounded to the nearest 10.

In general, in math or science, if the number is so big (or small) that you find yourself getting confused by all the 0s, you should use scientific notation. In science, use significant figures for rounding (unless your class doesn't use them, in which case you probably just round to two or three decimal places). In math, only round if they tell you to, otherwise try to use exact answers whenever possible. And never, ever round until the end of the problem, no matter what.
• my brain is going to die because of math 😭
• In the earlier videos, Sal went through each step of multiplying by decimals. The video "Scientific Notation Example 2" shows how to do that, and then how to adjust the result to be real scientific notation. By the time you get to this video, he figures that you have learned those basics.
• I am lost in the brain
• Wouldn't it be 10 to the power -10 on the first problem bc if you move to the left its dividing (negative exponent) but at he added 10 to the first power to 10 to the -9th power
• The 10^(-9) comes from multiplying:
10^8 * 10^(-12) * 10^(-5)
When we multiply a common base, we add the exponents.
8+(-12)+(-5) = -4+(-5) = -9
So, the new result is 10^(-9).

After multiplying the constants, he has:
40.1534 x 10^(-9)

Since the constant term is too large to be in proper scientific notation, it needs to be fixed. The number 40 is larger than 10. The decimal point needs to be moved 1 unit to the left, which creates:
4.01534 x 10^1 x 10^(-9)
It needs to be 10^1 because you would need to multiply 4.01534 by 10 to get back to 40.1534.

Then, again Sal adds the exponents on the 10s to get:
4.01534 x 10^(-8)

Hope this helps.
• Is ((7^3*2)+(4)^3-(1489*10^500)*(2652+16/2)/(23164+156))^2 relavent?
• Ok, so after like 30 minutes,(because I'm obsessed with math & solving weird problems XP) I found the approximate answer. BTW I'm showing my steps.

((7^3*2)+4^3-(1489*10^500)*(2652+16/2)/23164+156))^2
((686)+(64)-(1489*10^500)*2660)/23320))^2
(750-(1489*10^500)*0.1140651)^2
562500-(1489*10^500)*0.013108
562500-(1.489e+503)*0.013108
-1.489e+503*0.013108 EQUALS
-19517812e+501
Finally, I realize now that the expression was over complicated and got easy really quickly.

*Thanks for the challenge!!*
• How to write scientific notation in standard form
• Here's an example:
2.46 × 10^5
Now let's take the problem apart:
2.46 × (10^5) =
2.46 × (10 × 10 × 10 × 10 × 10) =
2.46 × (100,000), which should be fairly easy to solve since we just move the decimal pt. in "2.46" to the right as many places as there are zeros in "100,000" (5 zeros):
2.46 × (100,000) = 246,000.

This is the basic mathematics behind converting scientific notation to standard form. However this takes a long time, and it's easy to make little mistakes. Here's a "shortcut:"

Notice that in the above problem that "10^5" equals "100,000," which has five zeros, which means that you move the decimal five places. Five is the exponent of the "10" that your multiplying by.
So when you see the number:
2.46 × 10^5,
you can simply move the decimal 5 places to the right.
In the number "3.5 × 10^3," you would move the decimal 3 places to the right. For "2.987 × 10^8," move it 8 places. You get the idea, move the decimal as many places as the power of ten.

As you probably know, you'll sometimes have numbers like "4.3 × 10^(-2)," where the exponent is negative. In this case you would do the same thing except that you would move the decimal point to the left. (Leaving you with "0.043" as an answer.) Just remember that positive numbers are on the right side of the number line and negative numbers are on the left.