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## 8th grade (Eureka Math/EngageNY)

### Unit 1: Lesson 2

Topic B: Magnitude and scientific notation- Scientific notation examples
- Scientific notation example: 0.0000000003457
- Scientific notation
- Multiplying in scientific notation example
- 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
- Scientific notation word problem: red blood cells
- Scientific notation word problem: U.S. national debt
- Scientific notation word problem: speed of light
- Scientific notation word problems
- Scientific notation review

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# Scientific notation example: 0.0000000003457

CCSS.Math:

Can you imagine if you had to do calculations with very, very small numbers? How would you handle all those zeros to the right of the decimal? Thank goodness for scientific notation! Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- Hi, I have a question, I was doing practice on Khan Academy site. There was a question 52 thousandths, which I have to turn in scientific notation, I answered 5.2x10^4 because I know 52000 has 3 zeros and I also add 2, so it gave me 10^4 but I was shocked it is incorrect, but why? Can anyone tell me?(19 votes)
- Well the answer will be
*5.2*10^-2*. Because you have written**thousandths**not**thousands**. Both are very different.**thousands**are on the left side of the decimals but**thousandths**will be on the right side of the decimal. By the was 52 thousandths will be*0.052*(5 votes)

- Is there any short variant for writing 100000000000000000005000000000000000 in scientific notation?(8 votes)
- Sorry Johnathan, but that is quite wrong. If we have a number greater than 10, we move the decimal point to the left until we have a number between 1 and 10. Then, we count the number of times we moved the decimal and write that as an exponent over a base of 10. Finally, we write our number multiplied by the power of 10. So basically, to write this stupendously big number 100,000,000,000,000,000,005,000,000,000,000,000 in scientific notation, you will have to first get to the quadrillions and
*then*think until the decillions.

So, for the answer, you will write:**1.00000000000000000005 * 10^35**

Note: the sign * is the multiplication sign.(2 votes)

- Is there any application for scientific notation?(2 votes)
- Yes, there is! When dealing with big and long numbers, it can really be a pain writing out all of them, this is when scientific notation helps out! For example it will be really hard to write 300000000000, so we write it like 3•10^11(12 votes)

- Is there an algorithm for this?(3 votes)
- Technically what Nathan said is true, what the video explained has an "algorithm" to find "this'. Nathan just restated... etc(3 votes)

- how do you convert an ending equation like 3.05 x 10^5 back to scientific notation though?(0 votes)
- Your number is already in scientific notation.

Do you mean how do you go back to standard notation? If yes, you just need to shift the decimal point. The 10^5 means you are multiplying by 5 instances of 10. Each 10 shifts the decimal point 1 place to the right (you get a bigger number). Since there are 5 10's, shift the decimal point 5 places to the right and you get 305000 or 305,000

If the exponent had been negative 5, then you are essentially dividing by 5 tens and the decimal point will shift left (creating a smaller number).

Hope this helps.(8 votes)

- Does it always have to be a decimal or can it be like 32 * 10^2 instead of 3.2 * 10^3?(2 votes)
- The number can never exceed 10, so it would be wrong to use 32 instead of 3.2. That being said, you can still have it as whole numbers that are less than ten. For example, 300 would turn into 3 * 10^2.(3 votes)

- Hey, i have a question. i understood whats the meaning of "thousandths" but there was something like that

19 hundred-thousandths

i tried to write it. firstly, i thought 19 hundred can be 190 or 0.19 then i took that 190.. so then, it had to be 0.190 cause of thousandths. when i looked the answer, seriously it was so complicated. english isnt my main language but in this video, its not mention these ^^molds. i want to learn how its changing when we adding something or when we remove some number from that.

thanks for read.(3 votes) - do you have a video with multiplying scientific notations?(2 votes)
- I always get super confused if I try and be fancy with multiplying scientific notations. What I always do is FIRST change both numbers to standard form, multiply them, THEN change the answer back to scientific notation.(2 votes)

- how is it neagative ten power(2 votes)
- Okay , here is a problem to try and help you,

0.3643 into scientific notation ,

0.3643 start at your decimal point and move it one place to the right , then you have 03.643 .

now you drop the zero because you dont need it ,

because you dident move it to the left and you went the wrong way on the number line you have to make negative (-). once you have finished moving the decimal point to the RIGHT then you put it into the correct form .

3.643 times 10^-1 ( negative one because you went wrong way on number line *intentional *

and one because there is one number behind it .

Hope this help's !(2 votes)

- Makes sense. I wonder if there is a good way to remember which way is which for negative and positive. Don't you think?(3 votes)

## Video transcript

Express 0.0000000003457
in scientific notation. So let's just remind
ourselves what it means to be in
scientific notation. Scientific notation will be some
number times some power of 10 where this number right here--
let me write it this way. It's going to be greater
than or equal to 1, and it's going to
be less than 10. So over here, what
we want to put here is what that leading
number is going to be. And in general,
you're going to look for the first non-zero digit. And this is the
number that you're going to want to start off with. This is the only number you're
going to want to put ahead of or I guess to the left
of the decimal point. So we could write
3.457, and it's going to be multiplied
by 10 to something. Now let's think about
what we're going to have to multiply it by. To go from 3.457 to this
very, very small number, from 3.457, to get
to this, you have to move the decimal
to the left a bunch. You have to add a bunch of
zeroes to the left of the 3. You have to keep moving the
decimal over to the left. To do that, we're
essentially making the number much
much, much smaller. So we're not going
to multiply it by a positive exponent of 10. We're going to multiply it
times a negative exponent of 10. The equivalent is
you're dividing by a positive exponent of 10. And so the best way
to think about it, when you move an
exponent one to the left, you're dividing by 10, which
is equivalent to multiplying by 10 to the negative 1 power. Let me give you example here. So if I have 1 times 10 is
clearly just equal to 10. 1 times 10 to the
negative 1, that's equal to 1 times 1/10,
which is equal to 1/10. 1 times-- and let me actually
write a decimal, which is equal to 0-- let me actually-- I
skipped a step right there. Let me add 1 times 10 to the 0,
so we have something natural. So this is one times
10 to the first. One times 10 to the 0
is equal to 1 times 1, which is equal to 1. 1 times 10 to the negative
1 is equal to 1/10, which is equal to 0.1. If I do 1 times 10
to the negative 2, 10 to the negative 2 is 1
over 10 squared or 1/100. So this is going to be
1/100, which is 0.01. What's happening here? When I raise it to
a negative 1 power, I've essentially
moved the decimal from to the right of the
1 to the left of the 1. I've moved it from
there to there. When I raise it
to the negative 2, I moved it two over to the left. So how many times are we
going to have to move it over to the left to get this
number right over here? So let's think about
how many zeroes we have. So we have to move it one time
just to get in front of the 3. And then we have to
move it that many more times to get all of the zeroes
in there so that we have to move it one
time to get the 3. So if we started
here, we're going to move 1, 2, 3, 4, 5,
6, 7, 8, 9, 10 times. So this is going to be 3.457
times 10 to the negative 10 power. Let me just rewrite it. So 3.457 times 10 to
the negative 10 power. So in general,
what you want to do is you want to find the
first non-zero number here. Remember, you want a number
here that's between 1 and 10. And it can be equal to 1, but
it has to be less than 10. 3.457 definitely fits that bill. It's between 1 and 10. And then you just want
to count the leading zeroes between the
decimal and that number and include the number
because that tells you how many times you have
to shift the decimal over to actually get
this number up here. And so we have to shift
this decimal 10 times to the left to get
this thing up here.