# Standard Form

https://www.youtube.com/watch?v=VWhtJRV29R4

[00:03] When we have really big or really small numbers, we can use something called standard form as an easier way to write them.
[00:14] So here I've got the average distance to the sun, which is 149.6 billion meters, and I'm gonna write that in standard form.
[00:25] Standard form is a number between one and ten, so a number between 1 and 10 times 10 to the power of something.
[00:36] So with 149.6 billion meters, my number between 1 and 10 is going to be 1.496.
[00:49] So I can write 1.496 times 10 to the power of something.
[00:56] And the question is how many times do I multiply 1.496 by 10 to get?
[01:04] 149.6 billion.
[01:07] So if I multiply it by 10 once, I'll get 14.96.
[01:13] Twice, 149.6.
[01:16] Three times for 1496 and so on.
[01:21] So it's one, two, three, four, five, six, seven, eight, nine, ten, eleven times.
[01:31] So instead of saying, instead of writing out the whole number, I can write 1.496 times 10 to the power of 11, and that is in standard form.
[01:44] How about the radius of an electron?
[01:47] So I'm gonna have a number between one and ten, which is going to be 2.818.
[01:56] It's going to be times 10 to a power.
[02:02] But if I've got 2.818.
[02:05] I actually want to divide it by 10 a number of times to get to my number.
[02:13] In standard form we don't write divide by 10.
[02:16] We always write times by 10, and if we're dividing we use a negative power.
[02:23] Because if you have 3 to the power of negative 2, that is the same as 1 over 3 squared.
[02:32] So a negative power in effect makes it a division.
[02:39] So how many times do I want to divide 2.818 by 10 to get my number?
[02:46] So I've got 2.818 divided by 10 once, twice, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 times.
[03:07] So I can say the radius of an electron is 2.818 times 10 to the power of minus 15 meters.
[03:21] Okay, so let's look at writing numbers in standard form.
[03:25] So these numbers are a bit smaller, so it should be a bit easier.
[03:31] So with 27,000, I want a number between 1 and 10, which is going to be 2.7 times 10 to the power of something.
[03:42] How many times do I multiply 2.7 by 10 to get 27,000?
[03:48] One, two, three, four times.
[03:54] How about three hundred and fifteen thousand?
[03:57] So we need a number between 1 and 10, which is 3.15 this time, times 10 to the power of what?
[04:07] So 1, 2, 3, 4, 5 times.
[04:12] So one five times ten five times is the same as three hundred and fifteen thousand.
[04:19] How about zero point zero zero four?
[04:23] Number between one and ten, that's four times 10 to the power.
[04:30] I actually want to divide by 10, so I'm going to use a negative power.
[04:34] And how many times do I have to divide 4 by 10 to get 0.004?
[04:41] And that's three times.
[04:45] And finally zero point zero zero zero six eight.
[04:51] So number between one and ten is six point eight times ten.
[04:57] I want to divide by ten, so we have a negative power.
[05:03] And how many times do I divide by ten?
[05:06] That will be four.
[05:08] There's a slight trick here, so if you count the number of zeros including the one before the decimal place, one, two, three, four, times ten to the power of minus four.
[05:16] One, two, three times ten to the power of minus three.
[05:25] That works for negative powers.
[05:28] Okay.
[05:29] Questions for you to try, so pause the video, give them a go, and press play when you're ready to carry on.
[05:40] Okay, so the first one we've got 2.5 million.
[05:43] And that is 2.5 times 10 to the power of 6.
[05:48] A million is always times ten to the power of six.
[05:54] And we've got eighty-six thousand one hundred.
[05:59] So the number between one and ten, eight point six one, and that's times ten to the power of four.
[06:09] Zero point zero two three.
[06:12] That's going to be two point three.
[06:15] Between one and ten, two point three times ten to the power of we want to divide it by 10 so it's a negative power.
[06:23] And it's 2 times 2.3 divided by 10 twice equals 0.023.
[06:32] And 0.00 a number between 1 and 10, 9.7 times 10 to the power of it's going to be a negative power.
[06:47] How many times do I divide nine point seven by ten to get zero point zero zero zero zero nine seven?
[06:54] And that will be five.
[07:00] Okay, this time we're going to do it the other way around.
[07:03] So we're going to convert numbers in standard form back to ordinary numbers.
[07:10] So the first one I've got 3.2 times 10 to the power of three.
[07:17] So three point two times ten times ten times ten.
[07:22] Three point two times ten is thirty two.
[07:25] Times ten again three hundred and twenty.
[07:29] And a third time three thousand two hundred.
[07:32] Times ten to the power of three is always a thousands, it's always thousands.
[07:39] Eight point seven times ten to the power of six.
[07:42] So times 10 once is 87.
[07:46] Twice 870.
[07:49] 3, 4, 5, 6.
[07:53] So that's eight million seven hundred thousand.
[07:55] And to power of six is always millions.
[08:04] So the next one four point one three times ten to the power of negative four.
[08:09] So four point one three.
[08:11] Divided by ten four times because it's a negative power divided by ten four times.
[08:18] So once is zero point four one three.
[08:21] Twice zero point zero four one three.
[08:25] Three times zero point zero zero four one three and four times zero point zero zero zero four one three.
[08:35] So the negative power, the number of zeros including the one before the decimal place, is the same as the power.
[08:45] So four zeros, power of negative four.
[08:49] So the same thing with the power of negative six.
[08:52] Nine point six times ten to the power of negative six.
[08:55] So one, two, three, four, five, six and then the nine six.
[09:03] So nine point six divided by ten one, two, three, four, five, six times.
[09:15] Okay, four questions for you to try.
[09:19] Pause the video, give them a go, and press play when you're ready to carry on.
[09:26] Okay, five point one times ten to the power of five.
[09:31] So, times ten once is fifty one.
[09:35] Two, three, four, five times.
[09:40] So, five hundred and ten thousand.
[09:44] Four point nine times ten to the power of seven.
[09:47] So, times ten once to be 49, and then six more times.
[09:52] One, two, three, four, five, six.
[09:57] So that's 49 million.
[10:01] Three point seven times ten to the power of minus five.
[10:05] So we're going to have one, two, three, four, five zeros, a three and a seven.
[10:11] And we can check that.
[10:13] So three point seven divided by ten one.
[10:17] Two.
[10:17] Three.
[10:18] Four five times.
[10:21] And finally eight point two five times ten to the power of negative seven.
[10:28] So eight point two five divided by ten seven times.
[10:32] So zero point.
[10:34] So one two three four five six seven.
[10:39] And then eight two five.
[10:42] And again let's check it so 8.25 divided by 10 once twice three four five six seven times.
[10:57] Okay, in this question we're going to look at adding numbers in standard form.
[11:00] So we've got 3.2 times 10 to the power of 5 plus 8.7 times 10 to the power of 6.
[11:10] I can't add these numbers together straight away.
[11:11] I can't look at adding 3.2 and 8.7 because they're different things.
[11:19] So 3.2 times 10 to the 5 times 10 to the five is hundred thousands.
[11:25] So I've got three point two hundred thousands.
[11:28] And eight point seven times ten to the six is eight point seven millions.
[11:33] I can't add hundred thousands to millions.
[11:36] I can only add hundred thousands to hundred thousands or I can add millions to millions.
[11:43] So one way of answering this is to change them into ordinary numbers and then add them together.
[11:51] So I've got 3.2 times 10 to the power of 5.
[11:56] So three point two times ten once is thirty two, two, three, four, five times, so three hundred and twenty thousand.
[12:09] Eight point seven times ten to the six, eight point seven times ten once is eighty-seven, then two, three, four, five, six times.
[12:20] So 8.7 million.
[12:23] So now I can add these up by lining up in columns the two numbers so I can add the hundred thousands to the hundred thousands, the ten thousands to the ten thousands, and so on.
[12:38] So I've got 8.7 million and I've got 320,000.
[12:49] So I can add these up.
[12:52] So zero plus zero is zero.
[12:56] Ten thousands zero plus two is two.
[13:00] Hundred thousand seven plus three is ten, so I'll put zero there, carry one over into the millions.
[13:07] And eight plus one is nine.
[13:10] So I've got my answer of nine point zero two million.
[13:14] And I can write this in standard form.
[13:17] So it's nine point zero two.
[13:21] Times ten.
[13:23] One, two, three, four, five.
[13:26] Six times.
[13:32] Okay, there's a question for you to try.
[13:35] So pause the video, give it a go, and press play when you're ready to carry on.
[13:45] Okay, so we'll write them as ordinary numbers, so 5.8 times 10 seven times.
[13:52] So once to 58.
[13:55] 2, 3, 4, 5, 6, 7.
[13:59] And 9.4 times ten six times.
[14:02] Once to 94.
[14:04] Two, three, four, five, six times.
[14:09] So let's line them up.
[14:12] So I've got 58 million.
[14:16] 58 and six zeroes.
[14:22] And then I've got five zeroes for my other number, so line them up and then 94.
[14:34] So let's add them together.
[14:38] So all the zeros plus zeros is zero.
[14:43] Zero plus four is four.
[14:45] Eight plus nine is seventeen.
[14:48] So carry over the one.
[14:50] And five plus one is six.
[14:54] So I've got six seven four zero zero.
[14:59] And if I write in standard form, I want a number between 1 and 10, which is 6.74 times 10.
[15:09] 1, 2, 3, four, five, six, seven times.
[15:19] Now we're going to look at multiplying numbers in standard form.
[15:24] So work out the value of 3 times 10 to the power of 4 times 4 times 10 to the power of 6.
[15:33] This is very very similar to an indices question.
[15:39] So imagine we've got three x to the power of four multiplied by four x to the power of six.
[15:50] So what we do here is we multiply the two numbers together and multiply the things with the same base together.
[15:58] So I'd have three times four is twelve and x to the power of four times x to the power of six.
[16:05] Four x's times together times by another six x's.
[16:10] We multiply we add the powers.
[16:14] So I have 12 x to the power of ten.
[16:17] So standard form is the same thing.
[16:21] So I can do three times four I can times the two numbers together.
[16:26] Which is twelve.
[16:29] And then I can times the two things with the same base together, so 10 to the power 4 times 10 to the power of 6 is 10 to the power of 10.
[16:44] So I've got 12 times 10 to the power of 10.
[16:48] Now it's correct as a number, but it is not in standard form.
[16:56] Standard form has got to be a number between 1 and 10, and 12 is too big, so we need to make it between 1 and 10 to give our answer in standard form.
[17:10] So what we can do is we can change 12 into 1.2 times 10.
[17:19] So 12 is the same as 1.2 times 10, and that is times 10.
[17:26] To the power of 10 still.
[17:28] So I've changed 12 into 1.2 times 10.
[17:34] So it's still times 10 to the power of 10.
[17:40] And that leaves gives me 1.2 times 10 times 10 to the power of 10.
[17:48] 10 times 10 to the power 10 is 10 times together 11 times so 10 to the power of 1 times 10 to the power of 10 is 10 to the power of 11.
[18:01] And that is in standard form.
[18:05] Okay, another one here.
[18:06] So I've got 2 times 7 times the numbers together.
[18:11] So 2 times 10 to the power of eight times seven times ten to the power of minus five.
[18:19] So I've got two times seven which is fourteen.
[18:23] So times numbers together and times the tens together.
[18:29] Ten to the power of eight times ten to the power of five.
[18:33] When we multiply, we add the powers.
[18:37] Eight plus minus five, or eight minus five.
[18:41] That's three.
[18:43] Again, it's not in standard form.
[18:46] 14 isn't between 1 and 10.
[18:49] So change 14 into 1.4, one point four, times ten.
[18:59] Times ten to the power of three.
[19:02] And ten to the power of one times ten to the power of three is ten to the power of four.
[19:13] Okay, two questions for you to try.
[19:16] Pause the video, give them a go, and press play when you're ready to carry on.
[19:26] Okay, so we're going to multiply the numbers together.
[19:30] So multiply the 6 and the 7.
[19:33] 6 times 7 is 42.
[19:37] And we multiply the tens together.
[19:39] 10 to the power of 3 times 10 to the power of 8 is 10 to the power of 3 plus 8 which is 11.
[19:50] It is not in standard form.
[19:53] I've got 42 which isn't between 1 and 10.
[19:57] So I'll change that to 4.2 times 10.
[20:02] And that's times 10 to the power of 11.
[20:05] Ten times ten to the power of one times ten to the power of eleven is ten to the power of twelve.
[20:16] And the second one, five times eight is forty.
[20:23] Ten to the power of negative four times ten square of eight, negative four plus eight.
[20:30] is four
[20:33] again it's not in standard form
[20:36] so i'm going to change 40 into 4 times
[20:39] 10
[20:43] so i've got
[20:44] 4 times 10 times 10 times 10 times ten
[20:48] times ten
[20:49] which is four
[20:50] times ten
[20:52] to the power of five
[20:59] okay now dividing numbers in standard
[21:02] form
[21:03] so the same as multiplying
[21:05] what we can do here we can treat it the
[21:07] same as
[21:08] if it was
[21:10] the indices question 8
[21:13] x to the power of 9
[21:15] divided by 2 x to the power of 6
[21:18] so we can divide the two numbers
[21:21] and then
[21:22] divide
[21:23] the indices
[21:24] so the things with the same base and
[21:26] when we divide
[21:28] we take away the powers
[21:31] so this will be 8 divided by 2 is 4
[21:34] and x to the power of 9 divided by x to
[21:37] the power of 6
[21:39] 9 take away 6 is 3
[21:42] so that's 4 x to the power of three
[21:46] so it's the same thing with my standard
[21:48] form
[21:49] eight divided by two
[21:51] is four
[21:54] and for my tens
[21:55] ten to the power of nine
[21:57] divided by ten to the power of six
[22:00] nine take away six is three
[22:03] which gives me four times ten to the
[22:06] power of three
[22:07] that is in standard form
[22:09] four is between one and 10 so i don't
[22:12] have to do anything to it
[22:15] for the next one i'm going to use a
[22:17] calculator
[22:19] so we've got 8.575
[22:23] times 10 to the power of 9
[22:25] divided by 2.45
[22:28] times 10 to the power of negative 4.
[22:31] so if i grab the calculator
[22:35] so there's ways we can write this in the
[22:37] calculator so we can use our fraction
[22:40] button
[22:41] and we can write 8.575
[22:47] now there is a standard form button
[22:49] down here
[22:50] which we it's better to use
[22:53] so if i click that it comes up with
[22:55] times 10
[22:56] and then we just write the power
[22:58] so that's 9 in this case
[23:02] and
[23:05] we can write 2.45 on the bottom
[23:08] times 10 and we write the power which is
[23:11] negative 4
[23:14] and there we go we've been given our
[23:15] answer
[23:16] already in standard form by the
[23:18] calculator so when we have a really big
[23:21] number
[23:22] or a small number the calculator
[23:24] automatically gives us the answer in
[23:27] standard form so we've got 3.5 times 10
[23:32] to the power of 13.
[23:40] let me just grab that calculator again
[23:42] and i
[23:43] want to show you
[23:44] what you can't do
[23:46] so
[23:48] for a division question
[23:51] if you're not using the standard form
[23:52] button
[23:53] so if i wrote this top one in 8 times 10
[23:56] to the power of 9 divided by 2 times 10
[23:59] to the power of 6
[24:01] if i wrote 8 times
[24:03] ten
[24:04] to the power of nine
[24:08] divided
[24:10] by two times ten
[24:12] to the power of six
[24:14] if i wrote it in the calculated like
[24:16] that i would get the wrong answer
[24:19] the calculator is going to do 8 times 10
[24:22] to the power of 9. it's going to work
[24:24] out what that is then it's going to
[24:25] divide it by 2
[24:27] then it's going to multiply that answer
[24:30] by 10 to the power of 6.
[24:32] so if you're not using the standard form
[24:34] button
[24:35] you would have had to write these
[24:37] numbers in brackets
[24:41] to get the right answer
[24:43] so if you write them in brackets
[24:46] you get four thousand which is the same
[24:49] as four times ten to the power of three
[24:52] if you didn't write them in brackets
[25:03] you get 4 times 10 to the power of 15
[25:06] because the calculator is just going to
[25:08] do this in order
[25:11] and it's going to get it wrong however
[25:14] if you did use the standard form button
[25:17] and wrote eight times ten to the power
[25:20] of nine
[25:21] divided by two times ten to the power of
[25:24] six
[25:25] you would get the correct answer because
[25:28] it keeps
[25:29] 8 times 10 to the power of 9 as one
[25:31] number
[25:33] and 2 times 10 to the power of 6 as
[25:35] another number
[25:40] and i want to show you one more thing on
[25:42] the calculator
[25:43] so if i just go back
[25:46] to
[25:47] writing
[25:49] um standard form as ordinary numbers
[25:53] so if you've got a calculator available
[25:55] to you
[25:56] sometimes the calculator will give you
[25:58] the correct answer so for 3.2 times 10
[26:02] to the power of 3
[26:04] 3.2 times 10 to the power of 3
[26:07] you can just type it in the calculator
[26:10] and it will convert it
[26:12] to an ordinary number for you
[26:16] let's try nine point six times ten to
[26:19] the power of minus six
[26:25] so that one it won't do it only wants to
[26:28] give you the number in standard form
[26:30] but in that case
[26:32] if you wrote 0.00096
[26:44] it will convert that from an ordinary
[26:46] number into standard form
[26:48] for you
[26:51] okay let's move on
[26:55] so there's two questions for you to try
[26:57] here on dividing numbers in standard
[26:59] form
[27:00] give them a go
[27:01] and press play when you're ready to
[27:03] carry on
[27:08] okay so the first one we've got six
[27:10] point three times ten to the power of
[27:12] seven
[27:13] divided by two point one times ten to
[27:15] the power of three
[27:17] so we divide our
[27:19] two numbers in front
[27:21] 6.3 divided by 2.1
[27:24] how many 2.1 is going to 6.3 that's 3
[27:30] and 10 to the power of 7 divided
[27:32] by 10 to the power 3
[27:35] so
[27:37] 7 take away 3 is 4
[27:39] so that's
[27:41] 3 times 10 to the power of 4
[27:45] and the second one
[27:47] one point three times ten to the power
[27:49] of negative three
[27:50] divided by five point two
[27:53] times ten to the power of negative six
[27:56] so i'm going to do it without the
[27:57] calculator
[27:58] so 1.3 divided by five point two
[28:03] that is where it's one quarter
[28:06] or zero point two five
[28:12] and ten to the power of negative three
[28:14] divided by 10 to the power of negative 6
[28:18] so we're dividing so we're taking away
[28:20] the powers
[28:21] so it's negative 3
[28:23] take away
[28:25] negative 6.
[28:27] when we take away a negative that's the
[28:29] same as adding so it's negative three
[28:32] plus six
[28:33] which is
[28:35] three
[28:37] so zero point two five
[28:39] times ten to the power of three
[28:42] is it in standard form
[28:44] it's not
[28:46] because standard form
[28:47] has got to be a number between 1 and 10.
[28:51] so
[28:52] here i've got 0.25 times 10 times 10
[28:56] times 10.
[28:57] if i steal one of these tens
[29:00] so take off one of the tens
[29:05] so change 10 to the power 3
[29:08] into 10 times 10 squared
[29:11] and do 0.25 times 10
[29:14] to give me 2.5
[29:17] times 10
[29:19] squared
[29:24] so if you did type it into the
[29:25] calculator it would have given you an
[29:27] answer of 250
[29:29] and you would have had to convert that
[29:31] back into standard form
[29:37] okay
[29:38] to finish off i've got six questions
[29:40] here pause the video give them a go and
[29:43] press play when you're ready to carry on
[29:50] okay so question one write in standard
[29:52] form so i've got
[29:55] 5.4 million
[29:58] and i need a number between one and ten
[30:02] so that's going to be 5.4
[30:05] and how many times do i multiply 5.4 by
[30:08] 10 to get this number
[30:10] that's one two three four five six times
[30:18] for the second one i've got 0.0005
[30:22] i'm going to write it in standard form
[30:24] again
[30:25] so the number between 1 and 10
[30:28] is going to be 5.4
[30:32] and it's times 10
[30:34] this time i'm going to divide 5.4 by 10.
[30:38] it's going to be a negative power
[30:40] and how many times
[30:42] so one two three four
[30:46] so 5.4 times 10 to the power of negative
[30:50] four
[30:52] question three
[30:54] right as an ordinary number
[30:56] six point seven times ten to the power
[30:59] of negative four
[31:01] so six point seven divided by 10 four
[31:05] times
[31:07] give me 0.000
[31:09] so four zeros in total
[31:11] and then a six and a seven
[31:14] and i can check that if i've got 6.7
[31:17] divided by 10 once
[31:19] twice
[31:19] three four times
[31:24] question four
[31:25] work out four point eight times ten to
[31:27] the power of six
[31:29] plus three times ten to the power of 5
[31:33] so i'm going to write these as ordinary
[31:35] numbers first
[31:37] so 4.8 times 10 to the 6
[31:40] 4.8 times 10 once
[31:43] twice
[31:45] four
[31:46] five six times
[31:49] plus three times ten to the power of
[31:52] five
[31:53] so three times ten one
[31:55] two three four five times
[31:59] so i'm going to add these numbers up i'm
[32:01] going to write them in columns
[32:04] so 4.8 million
[32:09] plus 300 000
[32:12] so the numbers are lined up
[32:15] add them together
[32:20] so eight plus three is eleven
[32:23] carry over the one four plus one is five
[32:26] so that gives me 5.1 million
[32:33] if i wanted that in standard form that's
[32:35] 5.1 times 10 to the power of 6.
[32:42] question five
[32:44] work out three times ten to the power of
[32:46] twelve
[32:47] times four times ten to the power of
[32:50] negative five
[32:51] so i multiply my numbers three times
[32:54] four is twelve
[32:57] and my indices
[32:59] say 10 to the power of 12
[33:02] times 10 to the power of negative 5
[33:05] 12 plus negative 5
[33:08] is
[33:09] so i've got 12 times 10 to the power of
[33:12] seven
[33:14] that's not in standard form so i'm going
[33:16] to change that
[33:18] i'm going to change 12 into 1.2 times
[33:20] 10.
[33:22] so i've got 1.2 times 10
[33:24] times 10 to the power of 7
[33:27] which is 1.2
[33:29] times 10
[33:30] to the power of 8
[33:36] and finally question six we're dividing
[33:40] so i've got three times ten to the power
[33:42] of nine divided by four times ten to the
[33:45] power of five
[33:47] three divided by four
[33:49] is three quarters or
[33:51] 0.75
[33:55] and my powers
[33:57] i've got 10 to the power of 9 divided by
[34:00] 10 to the power of 5
[34:02] 9 take away 5 is 4
[34:04] so 0.75
[34:06] times 10 to the power of 4
[34:09] that's not in standard form
[34:11] because standard form is a number
[34:13] between 1 and 10
[34:15] so i'm going to steal one of the tens
[34:17] off of here
[34:18] so
[34:19] 0.75
[34:21] times 10
[34:23] times 10 to the power of 3
[34:25] and 0.75 times 10
[34:28] is 7.5
[34:30] so it's 7.5
[34:33] times ten
[34:34] to the power of three
[34:54] you