0:00:00.590,0:00:03.200
We're asked to[br]multiply 1.45 times 10

0:00:03.200,0:00:06.900
to the eighth times 9.2 times[br]10 to the negative 12th times

0:00:06.900,0:00:08.880
3.01 times 10 to[br]the negative fifth

0:00:08.880,0:00:11.920
and express the product in[br]both decimal and scientific

0:00:11.920,0:00:13.040
notation.

0:00:13.040,0:00:19.760
So this is 1.45 times 10[br]to the eighth power times--

0:00:19.760,0:00:22.374
and I could just write the[br]parentheses again like this,

0:00:22.374,0:00:24.790
but I'm just going to write[br]it as another multiplication--

0:00:24.790,0:00:29.970
times 9.2 times 10[br]to the negative 12th

0:00:29.970,0:00:37.620
and then times 3.01 times[br]10 to the negative fifth.

0:00:37.620,0:00:40.722
All this meant, when I wrote[br]these parentheses times next

0:00:40.722,0:00:42.430
to each other, I'm[br]just going to multiply

0:00:42.430,0:00:44.010
this expression[br]times this expression

0:00:44.010,0:00:45.330
times this expression.

0:00:45.330,0:00:47.590
And since everything is[br]involved multiplication,

0:00:47.590,0:00:51.020
it actually doesn't matter[br]what order I multiply in.

0:00:51.020,0:00:53.470
And so with that in mind,[br]I can swap the order here.

0:00:53.470,0:00:57.390
This is going to be the[br]same thing as 1.45-- that's

0:00:57.390,0:01:09.530
that right there-- times[br]9.2 times 3.01 times

0:01:09.530,0:01:11.370
10 to the eighth--[br]let me do that

0:01:11.370,0:01:18.490
in that purple color-- times[br]10 to the eighth times 10

0:01:18.490,0:01:24.910
to the negative 12th power times[br]10 to the negative fifth power.

0:01:28.880,0:01:30.750
And this is useful[br]because now I have

0:01:30.750,0:01:32.850
all of my powers of[br]10 right over here.

0:01:32.850,0:01:34.720
I could put parentheses[br]around that.

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

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And so I can simplify it.

0:01:39.730,0:01:42.750
If I have the same base[br]10 right over here,

0:01:42.750,0:01:44.620
so I can add the exponents.

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

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And then all of this[br]on the left-hand side--

0:01:57.160,0:02:02.187
let me get a calculator[br]out-- I have 1.45.

0:02:02.187,0:02:04.520
You could do it by hand, but[br]this is a little bit faster

0:02:04.520,0:02:08.850
and less likely to make a[br]careless mistake-- times 9.2

0:02:08.850,0:02:17.620
times 3.01, which[br]is equal to 40.1534.

0:02:17.620,0:02:22.380
So this is equal to 40.1534.

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

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to this thing.

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And so if we simplify[br]this exponent,

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you get 40.1534 times[br]10 to the 8 minus 12

0:02:33.630,0:02:36.520
is negative 4, minus[br]5 is negative 9.

0:02:36.520,0:02:39.180
10 to the negative 9 power.

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Now you might be tempted[br]to say that this is already

0:02:41.470,0:02:44.150
in scientific notation because[br]I have some number here

0:02:44.150,0:02:45.950
times some power of 10.

0:02:45.950,0:02:49.820
But this is not quite[br]official scientific notation.

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And that's because[br]in order for it

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to be in scientific notation,[br]this number right over here

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

0:03:01.060,0:03:03.420
And this is, obviously,[br]not less than 10.

0:03:03.420,0:03:05.860
Essentially, for it to be[br]in scientific notation,

0:03:05.860,0:03:08.495
you want a non-zero[br]digit right over here.

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And then you want[br]your decimal and then

0:03:10.120,0:03:11.850
the rest of everything else.

0:03:11.850,0:03:15.710
So here-- and you want[br]a non-zero single digit

0:03:15.710,0:03:16.210
over here.

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Here we obviously[br]have two digits.

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

0:03:22.910,0:03:25.280
You want this thing[br]to be less than 10

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and greater than or equal to 1.

0:03:27.959,0:03:30.000
So the best way to do that[br]is to write this thing

0:03:30.000,0:03:32.260
right over here in[br]scientific notation.

0:03:32.260,0:03:40.200
This is the same thing[br]as 4.01534 times 10.

0:03:40.200,0:03:43.480
And one way to think about[br]it is to go from 40 to 4,

0:03:43.480,0:03:46.990
we have to move this[br]decimal over to the left.

0:03:46.990,0:03:49.400
Moving a decimal over to[br]the left to go from 40 to 4

0:03:49.400,0:03:50.580
you're dividing by 10.

0:03:50.580,0:03:53.470
So you have to multiply by[br]10 so it all equals out.

0:03:53.470,0:03:55.539
Divide by 10 and[br]then multiply by 10.

0:03:55.539,0:03:58.080
Or another way to write it, or[br]another way to think about it,

0:03:58.080,0:04:03.820
is 4.0 and all this stuff times[br]10 is going to be 40.1534.

0:04:03.820,0:04:06.260
And so you're going to have[br]4-- all of this times 10

0:04:06.260,0:04:11.120
to the first power, that's[br]the same thing as 10-- times

0:04:11.120,0:04:15.070
this thing-- times 10 to[br]the negative ninth power.

0:04:15.070,0:04:17.640
And then once[br]again, powers of 10,

0:04:17.640,0:04:19.890
so it's 10 to the first[br]times 10 to the negative 9

0:04:19.890,0:04:24.800
is going to be 10 to the[br]negative eighth power.

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

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And now we have written[br]it in scientific notation.

0:04:38.490,0:04:39.865
Now, they wanted[br]us to express it

0:04:39.865,0:04:42.570
in both decimal and[br]scientific notation.

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

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they essentially want us to[br]multiply this out, expand this

0:04:48.810,0:04:49.610
out.

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And so the way to think about[br]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[br]looking at this number,

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I start with the[br]decimal right over here.

0:05:02.420,0:05:08.360
Now, every time I divide by[br]10, or if I multiply by 10

0:05:08.360,0:05:12.520
to the negative 1, I'm moving[br]this over to the left one spot.

0:05:12.520,0:05:15.290
So 10 to the negative[br]1-- if I multiply by 10

0:05:15.290,0:05:18.630
to the negative 1, that's the[br]same thing as dividing by 10.

0:05:18.630,0:05:21.440
And so I'm moving the[br]decimal over to the left one.

0:05:21.440,0:05:24.490
Here I'm multiplying by[br]10 to the negative 8.

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

0:05:27.960,0:05:30.901
So I'm going to want to move[br]the decimal to the left eight

0:05:30.901,0:05:31.400
times.

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And one way to[br]remember it-- look,

0:05:43.165,0:05:46.340
this is a very, very,[br]very, very small number.

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If I multiply this, I[br]should get a smaller number.

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So I should be moving[br]the decimal to the left.

0:05:51.360,0:05:53.510
If this was a[br]positive 8, then this

0:05:53.510,0:05:55.440
would be a very large number.

0:05:55.440,0:05:57.510
And so if I multiply[br]by a large power of 10,

0:05:57.510,0:05:59.510
I'm going to be moving[br]the decimal to the right.

0:05:59.510,0:06:02.790
So this whole thing[br]should evaluate

0:06:02.790,0:06:06.960
to being smaller than 4.01534.

0:06:06.960,0:06:11.010
So I move the decimal[br]eight times to the left.

0:06:11.010,0:06:13.770
I move it one time to the left[br]to get it right over here.

0:06:13.770,0:06:16.890
And then the next seven times,[br]I'm just going to add 0's.

0:06:16.890,0:06:22.710
So one, two, three, four,[br]five, six, seven 0's.

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

0:06:25.510,0:06:29.020
So now I notice, if you include[br]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[br]this digit gives us eight.

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

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The best way to[br]think about it is,

0:06:42.910,0:06:44.630
I started with the[br]decimal right here.

0:06:44.630,0:06:50.510
I moved once, twice, three,[br]four, five, six, seven,

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eight times.

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

0:06:54.730,0:06:57.060
And I get this number[br]right over here.

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And when you see a[br]number like this,

0:06:58.560,0:07:00.720
you start to appreciate[br]why we rewrite things

0:07:00.720,0:07:02.920
in scientific notation.

0:07:02.920,0:07:06.090
This is much easier to-- it[br]takes less space to write

0:07:06.090,0:07:09.020
and you immediately know[br]roughly how big this number is.

0:07:09.020,0:07:10.810
This is much harder to write.

0:07:10.810,0:07:12.420
You might even[br]forget a 0 when you

0:07:12.420,0:07:14.400
write it or you might add a 0.

0:07:14.400,0:07:17.410
And now the person has to sit[br]and count the 0's to figure out

0:07:17.410,0:07:20.950
essentially how large--or get[br]a rough sense of how large this

0:07:20.950,0:07:21.450
thing is.

0:07:21.450,0:07:24.660
It's one, two, three, four,[br]five, six, seven 0's, and you

0:07:24.660,0:07:25.940
have this digit right here.

0:07:25.940,0:07:27.440
That's what gets[br]us to that eight.

0:07:27.440,0:07:31.250
But this is a much, much more[br]complicated-looking number

0:07:31.250,0:07:34.076
than the one in[br]scientific notation.