WEBVTT

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We're asked to
multiply 1.45 times 10

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to the eighth times 9.2 times
10 to the negative 12th times

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3.01 times 10 to
the negative fifth

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and express the product in
both decimal and scientific

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notation.

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So this is 1.45 times 10
to the eighth power times--

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and I could just write the
parentheses again like this,

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but I'm just going to write
it as another multiplication--

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times 9.2 times 10
to the negative 12th

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and then times 3.01 times
10 to the negative fifth.

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All this meant, when I wrote
these parentheses times next

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to each other, I'm
just going to multiply

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this expression
times this expression

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times this expression.

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And since everything is
involved multiplication,

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it actually doesn't matter
what order I multiply in.

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And so with that in mind,
I can swap the order here.

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This is going to be the
same thing as 1.45-- that's

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that right there-- times
9.2 times 3.01 times

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10 to the eighth--
let me do that

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in that purple color-- times
10 to the eighth times 10

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to the negative 12th power times
10 to the negative fifth power.

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And this is useful
because now I have

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all of my powers of
10 right over here.

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I could put parentheses
around that.

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And I have all my non-powers
of 10 right over there.

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And so I can simplify it.

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If I have the same base
10 right over here,

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so I can add the exponents.

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This is going to be 10 to
the 8 minus 12 minus 5 power.

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And then all of this
on the left-hand side--

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let me get a calculator
out-- I have 1.45.

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You could do it by hand, but
this is a little bit faster

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and less likely to make a
careless mistake-- times 9.2

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times 3.01, which
is equal to 40.1534.

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So this is equal to 40.1534.

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And of course, this is going
to be multiplied times 10

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to this thing.

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And so if we simplify
this exponent,

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you get 40.1534 times
10 to the 8 minus 12

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is negative 4, minus
5 is negative 9.

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10 to the negative 9 power.

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Now you might be tempted
to say that this is already

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in scientific notation because
I have some number here

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times some power of 10.

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But this is not quite
official scientific notation.

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And that's because
in order for it

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to be in scientific notation,
this number right over here

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has to be greater than or
equal to 1 and less than 10.

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And this is, obviously,
not less than 10.

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Essentially, for it to be
in scientific notation,

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you want a non-zero
digit right over here.

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And then you want
your decimal and then

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the rest of everything else.

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So here-- and you want
a non-zero single digit

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over here.

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Here we obviously
have two digits.

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This is larger than 10-- or this
is greater than or equal to 10.

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You want this thing
to be less than 10

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and greater than or equal to 1.

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So the best way to do that
is to write this thing

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right over here in
scientific notation.

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This is the same thing
as 4.01534 times 10.

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And one way to think about
it is to go from 40 to 4,

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we have to move this
decimal over to the left.

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Moving a decimal over to
the left to go from 40 to 4

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you're dividing by 10.

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So you have to multiply by
10 so it all equals out.

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Divide by 10 and
then multiply by 10.

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Or another way to write it, or
another way to think about it,

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is 4.0 and all this stuff times
10 is going to be 40.1534.

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And so you're going to have
4-- all of this times 10

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to the first power, that's
the same thing as 10-- times

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this thing-- times 10 to
the negative ninth power.

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And then once
again, powers of 10,

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so it's 10 to the first
times 10 to the negative 9

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is going to be 10 to the
negative eighth power.

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And we still have this 4.01534
times 10 to the negative 8.

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And now we have written
it in scientific notation.

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Now, they wanted
us to express it

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in both decimal and
scientific notation.

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And when they're asking us to
write it in decimal notation,

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they essentially want us to
multiply this out, expand this

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out.

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And so the way to think about
it-- write these digits out.

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So I have 4, 0, 1, 5, 3, 4.

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And if I'm just
looking at this number,

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I start with the
decimal right over here.

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Now, every time I divide by
10, or if I multiply by 10

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to the negative 1, I'm moving
this over to the left one spot.

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So 10 to the negative
1-- if I multiply by 10

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to the negative 1, that's the
same thing as dividing by 10.

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And so I'm moving the
decimal over to the left one.

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Here I'm multiplying by
10 to the negative 8.

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Or you could say I'm dividing
by 10 to the eighth power.

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So I'm going to want to move
the decimal to the left eight

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times.

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And one way to
remember it-- look,

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this is a very, very,
very, very small number.

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If I multiply this, I
should get a smaller number.

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So I should be moving
the decimal to the left.

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If this was a
positive 8, then this

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would be a very large number.

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And so if I multiply
by a large power of 10,

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I'm going to be moving
the decimal to the right.

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So this whole thing
should evaluate

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to being smaller than 4.01534.

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So I move the decimal
eight times to the left.

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I move it one time to the left
to get it right over here.

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And then the next seven times,
I'm just going to add 0's.

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So one, two, three, four,
five, six, seven 0's.

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And I'll put a 0 in front of
the decimal just to clarify it.

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So now I notice, if you include
this digit right over here,

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I have a total of eight digits.

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I have seven 0's, and
this digit gives us eight.

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So again, one, two, three,
four, five, six, seven, eight.

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The best way to
think about it is,

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I started with the
decimal right here.

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I moved once, twice, three,
four, five, six, seven,

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eight times.

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That's what multiplying times
10 to the negative 8 did for us.

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And I get this number
right over here.

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And when you see a
number like this,

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you start to appreciate
why we rewrite things

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in scientific notation.

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This is much easier to-- it
takes less space to write

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and you immediately know
roughly how big this number is.

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This is much harder to write.

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You might even
forget a 0 when you

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write it or you might add a 0.

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And now the person has to sit
and count the 0's to figure out

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essentially how large--or get
a rough sense of how large this

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thing is.

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It's one, two, three, four,
five, six, seven 0's, and you

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have this digit right here.

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That's what gets
us to that eight.

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But this is a much, much more
complicated-looking number

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than the one in
scientific notation.

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