[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:00.59,Default,,0000,0000,0000,,
Dialogue: 0,0:00:00.59,0:00:03.20,Default,,0000,0000,0000,,We're asked to\Nmultiply 1.45 times 10
Dialogue: 0,0:00:03.20,0:00:06.90,Default,,0000,0000,0000,,to the eighth times 9.2 times\N10 to the negative 12th times
Dialogue: 0,0:00:06.90,0:00:08.88,Default,,0000,0000,0000,,3.01 times 10 to\Nthe negative fifth
Dialogue: 0,0:00:08.88,0:00:11.92,Default,,0000,0000,0000,,and express the product in\Nboth decimal and scientific
Dialogue: 0,0:00:11.92,0:00:13.04,Default,,0000,0000,0000,,notation.
Dialogue: 0,0:00:13.04,0:00:19.76,Default,,0000,0000,0000,,So this is 1.45 times 10\Nto the eighth power times--
Dialogue: 0,0:00:19.76,0:00:22.37,Default,,0000,0000,0000,,and I could just write the\Nparentheses again like this,
Dialogue: 0,0:00:22.37,0:00:24.79,Default,,0000,0000,0000,,but I'm just going to write\Nit as another multiplication--
Dialogue: 0,0:00:24.79,0:00:29.97,Default,,0000,0000,0000,,times 9.2 times 10\Nto the negative 12th
Dialogue: 0,0:00:29.97,0:00:37.62,Default,,0000,0000,0000,,and then times 3.01 times\N10 to the negative fifth.
Dialogue: 0,0:00:37.62,0:00:40.72,Default,,0000,0000,0000,,All this meant, when I wrote\Nthese parentheses times next
Dialogue: 0,0:00:40.72,0:00:42.43,Default,,0000,0000,0000,,to each other, I'm\Njust going to multiply
Dialogue: 0,0:00:42.43,0:00:44.01,Default,,0000,0000,0000,,this expression\Ntimes this expression
Dialogue: 0,0:00:44.01,0:00:45.33,Default,,0000,0000,0000,,times this expression.
Dialogue: 0,0:00:45.33,0:00:47.59,Default,,0000,0000,0000,,And since everything is\Ninvolved multiplication,
Dialogue: 0,0:00:47.59,0:00:51.02,Default,,0000,0000,0000,,it actually doesn't matter\Nwhat order I multiply in.
Dialogue: 0,0:00:51.02,0:00:53.47,Default,,0000,0000,0000,,And so with that in mind,\NI can swap the order here.
Dialogue: 0,0:00:53.47,0:00:57.39,Default,,0000,0000,0000,,This is going to be the\Nsame thing as 1.45-- that's
Dialogue: 0,0:00:57.39,0:01:09.53,Default,,0000,0000,0000,,that right there-- times\N9.2 times 3.01 times
Dialogue: 0,0:01:09.53,0:01:11.37,Default,,0000,0000,0000,,10 to the eighth--\Nlet me do that
Dialogue: 0,0:01:11.37,0:01:18.49,Default,,0000,0000,0000,,in that purple color-- times\N10 to the eighth times 10
Dialogue: 0,0:01:18.49,0:01:24.91,Default,,0000,0000,0000,,to the negative 12th power times\N10 to the negative fifth power.
Dialogue: 0,0:01:24.91,0:01:28.88,Default,,0000,0000,0000,,
Dialogue: 0,0:01:28.88,0:01:30.75,Default,,0000,0000,0000,,And this is useful\Nbecause now I have
Dialogue: 0,0:01:30.75,0:01:32.85,Default,,0000,0000,0000,,all of my powers of\N10 right over here.
Dialogue: 0,0:01:32.85,0:01:34.72,Default,,0000,0000,0000,,I could put parentheses\Naround that.
Dialogue: 0,0:01:34.72,0:01:38.46,Default,,0000,0000,0000,,And I have all my non-powers\Nof 10 right over there.
Dialogue: 0,0:01:38.46,0:01:39.73,Default,,0000,0000,0000,,And so I can simplify it.
Dialogue: 0,0:01:39.73,0:01:42.75,Default,,0000,0000,0000,,If I have the same base\N10 right over here,
Dialogue: 0,0:01:42.75,0:01:44.62,Default,,0000,0000,0000,,so I can add the exponents.
Dialogue: 0,0:01:44.62,0:01:52.32,Default,,0000,0000,0000,,This is going to be 10 to\Nthe 8 minus 12 minus 5 power.
Dialogue: 0,0:01:52.32,0:01:54.96,Default,,0000,0000,0000,,
Dialogue: 0,0:01:54.96,0:01:57.16,Default,,0000,0000,0000,,And then all of this\Non the left-hand side--
Dialogue: 0,0:01:57.16,0:02:02.19,Default,,0000,0000,0000,,let me get a calculator\Nout-- I have 1.45.
Dialogue: 0,0:02:02.19,0:02:04.52,Default,,0000,0000,0000,,You could do it by hand, but\Nthis is a little bit faster
Dialogue: 0,0:02:04.52,0:02:08.85,Default,,0000,0000,0000,,and less likely to make a\Ncareless mistake-- times 9.2
Dialogue: 0,0:02:08.85,0:02:17.62,Default,,0000,0000,0000,,times 3.01, which\Nis equal to 40.1534.
Dialogue: 0,0:02:17.62,0:02:22.38,Default,,0000,0000,0000,,So this is equal to 40.1534.
Dialogue: 0,0:02:22.38,0:02:24.81,Default,,0000,0000,0000,,And of course, this is going\Nto be multiplied times 10
Dialogue: 0,0:02:24.81,0:02:25.95,Default,,0000,0000,0000,,to this thing.
Dialogue: 0,0:02:25.95,0:02:27.71,Default,,0000,0000,0000,,And so if we simplify\Nthis exponent,
Dialogue: 0,0:02:27.71,0:02:33.63,Default,,0000,0000,0000,,you get 40.1534 times\N10 to the 8 minus 12
Dialogue: 0,0:02:33.63,0:02:36.52,Default,,0000,0000,0000,,is negative 4, minus\N5 is negative 9.
Dialogue: 0,0:02:36.52,0:02:39.18,Default,,0000,0000,0000,,10 to the negative 9 power.
Dialogue: 0,0:02:39.18,0:02:41.47,Default,,0000,0000,0000,,Now you might be tempted\Nto say that this is already
Dialogue: 0,0:02:41.47,0:02:44.15,Default,,0000,0000,0000,,in scientific notation because\NI have some number here
Dialogue: 0,0:02:44.15,0:02:45.95,Default,,0000,0000,0000,,times some power of 10.
Dialogue: 0,0:02:45.95,0:02:49.82,Default,,0000,0000,0000,,But this is not quite\Nofficial scientific notation.
Dialogue: 0,0:02:49.82,0:02:51.53,Default,,0000,0000,0000,,And that's because\Nin order for it
Dialogue: 0,0:02:51.53,0:02:55.70,Default,,0000,0000,0000,,to be in scientific notation,\Nthis number right over here
Dialogue: 0,0:02:55.70,0:03:01.06,Default,,0000,0000,0000,,has to be greater than or\Nequal to 1 and less than 10.
Dialogue: 0,0:03:01.06,0:03:03.42,Default,,0000,0000,0000,,And this is, obviously,\Nnot less than 10.
Dialogue: 0,0:03:03.42,0:03:05.86,Default,,0000,0000,0000,,Essentially, for it to be\Nin scientific notation,
Dialogue: 0,0:03:05.86,0:03:08.50,Default,,0000,0000,0000,,you want a non-zero\Ndigit right over here.
Dialogue: 0,0:03:08.50,0:03:10.12,Default,,0000,0000,0000,,And then you want\Nyour decimal and then
Dialogue: 0,0:03:10.12,0:03:11.85,Default,,0000,0000,0000,,the rest of everything else.
Dialogue: 0,0:03:11.85,0:03:15.71,Default,,0000,0000,0000,,So here-- and you want\Na non-zero single digit
Dialogue: 0,0:03:15.71,0:03:16.21,Default,,0000,0000,0000,,over here.
Dialogue: 0,0:03:16.21,0:03:18.93,Default,,0000,0000,0000,,Here we obviously\Nhave two digits.
Dialogue: 0,0:03:18.93,0:03:22.91,Default,,0000,0000,0000,,This is larger than 10-- or this\Nis greater than or equal to 10.
Dialogue: 0,0:03:22.91,0:03:25.28,Default,,0000,0000,0000,,You want this thing\Nto be less than 10
Dialogue: 0,0:03:25.28,0:03:27.96,Default,,0000,0000,0000,,and greater than or equal to 1.
Dialogue: 0,0:03:27.96,0:03:30.00,Default,,0000,0000,0000,,So the best way to do that\Nis to write this thing
Dialogue: 0,0:03:30.00,0:03:32.26,Default,,0000,0000,0000,,right over here in\Nscientific notation.
Dialogue: 0,0:03:32.26,0:03:40.20,Default,,0000,0000,0000,,This is the same thing\Nas 4.01534 times 10.
Dialogue: 0,0:03:40.20,0:03:43.48,Default,,0000,0000,0000,,And one way to think about\Nit is to go from 40 to 4,
Dialogue: 0,0:03:43.48,0:03:46.99,Default,,0000,0000,0000,,we have to move this\Ndecimal over to the left.
Dialogue: 0,0:03:46.99,0:03:49.40,Default,,0000,0000,0000,,Moving a decimal over to\Nthe left to go from 40 to 4
Dialogue: 0,0:03:49.40,0:03:50.58,Default,,0000,0000,0000,,you're dividing by 10.
Dialogue: 0,0:03:50.58,0:03:53.47,Default,,0000,0000,0000,,So you have to multiply by\N10 so it all equals out.
Dialogue: 0,0:03:53.47,0:03:55.54,Default,,0000,0000,0000,,Divide by 10 and\Nthen multiply by 10.
Dialogue: 0,0:03:55.54,0:03:58.08,Default,,0000,0000,0000,,Or another way to write it, or\Nanother way to think about it,
Dialogue: 0,0:03:58.08,0:04:03.82,Default,,0000,0000,0000,,is 4.0 and all this stuff times\N10 is going to be 40.1534.
Dialogue: 0,0:04:03.82,0:04:06.26,Default,,0000,0000,0000,,And so you're going to have\N4-- all of this times 10
Dialogue: 0,0:04:06.26,0:04:11.12,Default,,0000,0000,0000,,to the first power, that's\Nthe same thing as 10-- times
Dialogue: 0,0:04:11.12,0:04:15.07,Default,,0000,0000,0000,,this thing-- times 10 to\Nthe negative ninth power.
Dialogue: 0,0:04:15.07,0:04:17.64,Default,,0000,0000,0000,,And then once\Nagain, powers of 10,
Dialogue: 0,0:04:17.64,0:04:19.89,Default,,0000,0000,0000,,so it's 10 to the first\Ntimes 10 to the negative 9
Dialogue: 0,0:04:19.89,0:04:24.80,Default,,0000,0000,0000,,is going to be 10 to the\Nnegative eighth power.
Dialogue: 0,0:04:24.80,0:04:31.78,Default,,0000,0000,0000,,And we still have this 4.01534\Ntimes 10 to the negative 8.
Dialogue: 0,0:04:31.78,0:04:38.49,Default,,0000,0000,0000,,And now we have written\Nit in scientific notation.
Dialogue: 0,0:04:38.49,0:04:39.86,Default,,0000,0000,0000,,Now, they wanted\Nus to express it
Dialogue: 0,0:04:39.86,0:04:42.57,Default,,0000,0000,0000,,in both decimal and\Nscientific notation.
Dialogue: 0,0:04:42.57,0:04:45.15,Default,,0000,0000,0000,,And when they're asking us to\Nwrite it in decimal notation,
Dialogue: 0,0:04:45.15,0:04:48.81,Default,,0000,0000,0000,,they essentially want us to\Nmultiply this out, expand this
Dialogue: 0,0:04:48.81,0:04:49.61,Default,,0000,0000,0000,,out.
Dialogue: 0,0:04:49.61,0:04:53.14,Default,,0000,0000,0000,,And so the way to think about\Nit-- write these digits out.
Dialogue: 0,0:04:53.14,0:04:58.02,Default,,0000,0000,0000,,So I have 4, 0, 1, 5, 3, 4.
Dialogue: 0,0:04:58.02,0:04:59.90,Default,,0000,0000,0000,,And if I'm just\Nlooking at this number,
Dialogue: 0,0:04:59.90,0:05:02.42,Default,,0000,0000,0000,,I start with the\Ndecimal right over here.
Dialogue: 0,0:05:02.42,0:05:08.36,Default,,0000,0000,0000,,Now, every time I divide by\N10, or if I multiply by 10
Dialogue: 0,0:05:08.36,0:05:12.52,Default,,0000,0000,0000,,to the negative 1, I'm moving\Nthis over to the left one spot.
Dialogue: 0,0:05:12.52,0:05:15.29,Default,,0000,0000,0000,,So 10 to the negative\N1-- if I multiply by 10
Dialogue: 0,0:05:15.29,0:05:18.63,Default,,0000,0000,0000,,to the negative 1, that's the\Nsame thing as dividing by 10.
Dialogue: 0,0:05:18.63,0:05:21.44,Default,,0000,0000,0000,,And so I'm moving the\Ndecimal over to the left one.
Dialogue: 0,0:05:21.44,0:05:24.49,Default,,0000,0000,0000,,Here I'm multiplying by\N10 to the negative 8.
Dialogue: 0,0:05:24.49,0:05:27.96,Default,,0000,0000,0000,,Or you could say I'm dividing\Nby 10 to the eighth power.
Dialogue: 0,0:05:27.96,0:05:30.90,Default,,0000,0000,0000,,So I'm going to want to move\Nthe decimal to the left eight
Dialogue: 0,0:05:30.90,0:05:31.40,Default,,0000,0000,0000,,times.
Dialogue: 0,0:05:31.40,0:05:41.75,Default,,0000,0000,0000,,
Dialogue: 0,0:05:41.75,0:05:43.16,Default,,0000,0000,0000,,And one way to\Nremember it-- look,
Dialogue: 0,0:05:43.16,0:05:46.34,Default,,0000,0000,0000,,this is a very, very,\Nvery, very small number.
Dialogue: 0,0:05:46.34,0:05:48.86,Default,,0000,0000,0000,,If I multiply this, I\Nshould get a smaller number.
Dialogue: 0,0:05:48.86,0:05:51.36,Default,,0000,0000,0000,,So I should be moving\Nthe decimal to the left.
Dialogue: 0,0:05:51.36,0:05:53.51,Default,,0000,0000,0000,,If this was a\Npositive 8, then this
Dialogue: 0,0:05:53.51,0:05:55.44,Default,,0000,0000,0000,,would be a very large number.
Dialogue: 0,0:05:55.44,0:05:57.51,Default,,0000,0000,0000,,And so if I multiply\Nby a large power of 10,
Dialogue: 0,0:05:57.51,0:05:59.51,Default,,0000,0000,0000,,I'm going to be moving\Nthe decimal to the right.
Dialogue: 0,0:05:59.51,0:06:02.79,Default,,0000,0000,0000,,So this whole thing\Nshould evaluate
Dialogue: 0,0:06:02.79,0:06:06.96,Default,,0000,0000,0000,,to being smaller than 4.01534.
Dialogue: 0,0:06:06.96,0:06:11.01,Default,,0000,0000,0000,,So I move the decimal\Neight times to the left.
Dialogue: 0,0:06:11.01,0:06:13.77,Default,,0000,0000,0000,,I move it one time to the left\Nto get it right over here.
Dialogue: 0,0:06:13.77,0:06:16.89,Default,,0000,0000,0000,,And then the next seven times,\NI'm just going to add 0's.
Dialogue: 0,0:06:16.89,0:06:22.71,Default,,0000,0000,0000,,So one, two, three, four,\Nfive, six, seven 0's.
Dialogue: 0,0:06:22.71,0:06:25.51,Default,,0000,0000,0000,,And I'll put a 0 in front of\Nthe decimal just to clarify it.
Dialogue: 0,0:06:25.51,0:06:29.02,Default,,0000,0000,0000,,So now I notice, if you include\Nthis digit right over here,
Dialogue: 0,0:06:29.02,0:06:30.91,Default,,0000,0000,0000,,I have a total of eight digits.
Dialogue: 0,0:06:30.91,0:06:33.93,Default,,0000,0000,0000,,
Dialogue: 0,0:06:33.93,0:06:36.84,Default,,0000,0000,0000,,I have seven 0's, and\Nthis digit gives us eight.
Dialogue: 0,0:06:36.84,0:06:41.49,Default,,0000,0000,0000,,So again, one, two, three,\Nfour, five, six, seven, eight.
Dialogue: 0,0:06:41.49,0:06:42.91,Default,,0000,0000,0000,,The best way to\Nthink about it is,
Dialogue: 0,0:06:42.91,0:06:44.63,Default,,0000,0000,0000,,I started with the\Ndecimal right here.
Dialogue: 0,0:06:44.63,0:06:50.51,Default,,0000,0000,0000,,I moved once, twice, three,\Nfour, five, six, seven,
Dialogue: 0,0:06:50.51,0:06:51.52,Default,,0000,0000,0000,,eight times.
Dialogue: 0,0:06:51.52,0:06:54.73,Default,,0000,0000,0000,,That's what multiplying times\N10 to the negative 8 did for us.
Dialogue: 0,0:06:54.73,0:06:57.06,Default,,0000,0000,0000,,And I get this number\Nright over here.
Dialogue: 0,0:06:57.06,0:06:58.56,Default,,0000,0000,0000,,And when you see a\Nnumber like this,
Dialogue: 0,0:06:58.56,0:07:00.72,Default,,0000,0000,0000,,you start to appreciate\Nwhy we rewrite things
Dialogue: 0,0:07:00.72,0:07:02.92,Default,,0000,0000,0000,,in scientific notation.
Dialogue: 0,0:07:02.92,0:07:06.09,Default,,0000,0000,0000,,This is much easier to-- it\Ntakes less space to write
Dialogue: 0,0:07:06.09,0:07:09.02,Default,,0000,0000,0000,,and you immediately know\Nroughly how big this number is.
Dialogue: 0,0:07:09.02,0:07:10.81,Default,,0000,0000,0000,,This is much harder to write.
Dialogue: 0,0:07:10.81,0:07:12.42,Default,,0000,0000,0000,,You might even\Nforget a 0 when you
Dialogue: 0,0:07:12.42,0:07:14.40,Default,,0000,0000,0000,,write it or you might add a 0.
Dialogue: 0,0:07:14.40,0:07:17.41,Default,,0000,0000,0000,,And now the person has to sit\Nand count the 0's to figure out
Dialogue: 0,0:07:17.41,0:07:20.95,Default,,0000,0000,0000,,essentially how large--or get\Na rough sense of how large this
Dialogue: 0,0:07:20.95,0:07:21.45,Default,,0000,0000,0000,,thing is.
Dialogue: 0,0:07:21.45,0:07:24.66,Default,,0000,0000,0000,,It's one, two, three, four,\Nfive, six, seven 0's, and you
Dialogue: 0,0:07:24.66,0:07:25.94,Default,,0000,0000,0000,,have this digit right here.
Dialogue: 0,0:07:25.94,0:07:27.44,Default,,0000,0000,0000,,That's what gets\Nus to that eight.
Dialogue: 0,0:07:27.44,0:07:31.25,Default,,0000,0000,0000,,But this is a much, much more\Ncomplicated-looking number
Dialogue: 0,0:07:31.25,0:07:34.08,Default,,0000,0000,0000,,than the one in\Nscientific notation.
Dialogue: 0,0:07:34.08,0:07:34.58,Default,,0000,0000,0000,,