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Studying for a test? Prepare with these 8 lessons on Exponents, radicals, and scientific notation.

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# Scientific notation examples

Video transcript

There are two whole Khan
Academy videos on what scientific notation is, why
we even worry about it. And it also goes through
a few examples. And what I want to do in this
video is just use a ck12.org Algebra I book to do some more
scientific notation examples. So let's take some things
that are written in scientific notation. Just as a reminder, scientific
notation is useful because it allows us to write really
large, or really small numbers, in ways that are easy
for our brains to, one, write down, and two, understand. So let's write down
some numbers. So let's say I have 3.102
times 10 to the second. And I want to write it as
just a numerical value. It's in scientific
notation already. It's written as a product
with a power of 10. So how do I write this? It's just a numeral. Well, there's a slow way
and the fast way. The slow way is to say, well,
this is the same thing as 3.102 times 100, which means if
you multiplied 3.102 times 100, it'll be 3, 1, 0, 2,
with two 0's behind it. And then we have 1, 2, 3 numbers
behind the decimal point, and that'd be
the right answer. This is equal to 310.2. Now, a faster way to do this
is just to say, well, look, right now I have only the 3 in
front of the decimal point. When I take something times 10
to the second power, I'm essentially shifting the decimal
point 2 to the right. So 3.102 times 10 to the second
power is the same thing as-- if I shift the decimal
point 1, and then 2, because this is 10 to the second
power-- it's same thing as 310.2. So this might be a faster
way of doing it. Every time you multiply it by
10, you shift the decimal to the right by 1. Let's do another example. Let's say I had 7.4 times
10 to the fourth. Well, let's just do
this the fast way. Let's shift the decimal
4 to the right. So 7.4 times 10 to the fourth. Times 10 to the 1, you're
going to get 74. Then times 10 to the second,
you're going to get 740. We're going to have to add a
0 there, because we have to shift the decimal again. 10 to the third, you're
going to have 7,400. And then 10 to the fourth,
you're going to have 74,000. Notice, I just took
this decimal and went 1, 2, 3, 4 spaces. So this is equal to 74,000. And when I had 74, and I had to
shift the decimal 1 more to the right, I had to
throw in a 0 here. I'm multiplying it by 10. Another way to think about it
is, I need 10 spaces between the leading digit
and the decimal. So right here, I only
have 1 space. I'll need 4 spaces,
So, 1, 2, 3, 4. Let's do a few more examples,
because I think the more examples, the more you'll
get what's going on. So I have 1.75 times 10
to the negative 3. This is in scientific notation,
and I want to just write the numerical
value of this. So when you take something to
the negative times 10 to the negative power, you shift
the decimal to the left. So this is 1.75. So if you do it times 10 to the
negative 1 power, you'll go 1 to the left. But if you do times 10 to the
negative 2 power, you'll go 2 to the left. And you'd have to
put a 0 here. And if you do times 10 to the
negative 3, you'd go 3 to the left, and you would have
to add another 0. So you take this decimal and
go 1, 2, 3 to the left. So our answer would be 0.00175
is the same thing as 1.75 times 10 to the negative 3. And another way to check that
you got the right answer is if you have a 1 right here, if you
count the 1, 1 including the 0's to the right of the
decimal should be the same as the negative exponent here. So you have 1, 2, 3 numbers
behind the decimal. That's the same thing as to
the negative 3 power. You're doing 1,000th, so this
is 1,000th right there. Let's do another example. Actually let's mix it up. Let's start with something
that's written as a numeral and then write it in scientific
notation. So let's say I have 120,000. So that's just its numerical
value, and I want to write it in scientific notation. So this I can write as-- I take
the leading digit-- 1.2 times 10 to the-- and I just
count how many digits there are behind the leading digit. 1, 2, 3, 4, 5. So 1.2 times 10 to the fifth. And if you want to internalize
why that makes sense, 10 to the fifth is 10,000. So 1.2-- 10 to the
fifth is 100,000. So it's 1.2 times--
1, 1, 2, 3, 4, 5. You have five 0's. That's 10 to the fifth. So 1.2 times 100,000 is
going to be a 120,000. It's going to be 1 and 1/5
times 100,000, so 120. So hopefully that's
sinking in. So let's do another one. Let's say the numerical
value is 1,765,244. I want to write this in
scientific notation, so I take the leading digit, 1,
put a decimal sign. Everything else goes
behind the decimal. 7, 6, 5, 2, 4, 4. And then you count how many
digits there were between the leading digit, and I guess, you
could imagine, the first decimal sign. Because you could have numbers
that keep going over here. So between the leading digit
and the decimal sign. And you have 1, 2, 3,
4, 5, 6 digits. So this is times 10
to the sixth. And 10 to the sixth
is just 1 million. So it's 1.765244 times 1
million, which makes sense. Roughly 1.7 times million
is roughly 1.7 million. This is a little bit
more than 1.7 million, so it makes sense. Let's do another one. How do I write 12 in scientific
notation? Same drill. It's equal to 1.2 times-- well,
we only have 1 digit between the 1 and the decimal
spot, or the decimal point. So it's 1.2 times 10 to the
first power, or 1.2 times 10, which is definitely
equal to 12. Let's do a couple of examples
where we're taking 10 to a negative power. So let's say we had 0.00281, and
we want to write this in scientific notation. So what you do, is you just have
to think, well, how many digits are there to include
the leading numeral in the value? So what I mean there
is count, 1, 2, 3. So what we want to
do is we move the decimal 1, 2, 3 spaces. So one way you could think about
it is, you can multiply. To move the decimal to the
right 3 spaces, you would multiply it by 10
to the third. But if you're multiplying
something by 10 to the third, you're changing its values. So we also have to multiply
by 10 to the negative 3. Only this way will you not
change the value, right? If I multiply by 10 to the 3,
times 10 to the negative 2-- 3 minus 3 is 0-- this is just
like multiplying it by 1. So what is this going
to equal? If I take the decimal and I move
it 3 spaces to the right, this part right here is going
to be equal to 2.81. And then we're left with
this one, times 10 to the negative 3. Now, a very quick way to do it
is just to say, look, let me count-- including the leading
numeral-- how many spaces I have behind the decimal. 1, 2, 3. So it's going to be 2.81
times 10 to the negative 1, 2, 3 power. Let's do one more like that. Let me actually scroll
up here. Let's do one more like that. Let's say I have 1, 2, 3, 4, 5,
6-- how many 0's do I have in this problem? Well, I'll just make
up something. 0, 2, 7. And you wanted to write that
in scientific notation. Well, you count all the
digits up to the 2, behind the decimal. So 1, 2, 3, 4, 5, 6, 7, 8. So this is going to
be 2.7 times 10 to the negative 8 power. Now let's do another one,
where we start with the scientific notation value
and we want to go to the numeric value. Just to mix things up. So let's say you have
2.9 times 10 to the negative fifth. So one way to think about is,
this leading numeral, plus all 0's to the left of the
decimal spot, is going to be five digits. So you have a 2 and a 9,
and then you're going to have 4 more 0's. 1, 2, 3, 4. And then you're going to
have your decimal. And how did I know 4 0's? Because I'm counting,, this is
1, 2, 3, 4, 5 spaces behind the decimal, including
the leading numeral. And so it's 0.000029. And just to verify, do
the other technique. How do I write this in
scientific notation? I count all of the digits, all
of the leading 0's behind the decimal, including the leading
non-zero numeral. So I have 1, 2, 3,
4, 5 digits. So it's 10 to the negative 5. And so it'll be 2.9 times
10 to the negative 5. And once again, this isn't
just some type of black magic here. This actually makes
a lot of sense. If I wanted to get this number
to 2.9, what I would have to do is move the decimal over 1,
2, 3, 4, 5 spots, like that. And to get the decimal to move
over the right by 5 spots-- let's just say with
0, 0, 0, 0, 2, 9. If I multiply it by 10 to the
fifth, I'm also going to have to multiply it by 10
to the negative 5. So I don't want to change
the number. This right here is just
multiplying something by 1. 10 to the fifth times 10
to the negative 5 is 1. So this right here is
essentially going to move the decimal 5 to the right. 1, 2, 3, 4, 5. So this will be 2.5, and then
we're going to be left with times 10 to the negative 5. Anyway, hopefully, you
found that scientific notation drill useful.