### Video Transcript

Part a) Write 3040000 in standard
form. Part b) Write 3.7 times 10 to the
power of negative three as an ordinary number.

So what we need to do in part a is
actually decide what is standard form. Well, standard form is actually
when a number is in the form where we have 𝑎 multiplied by 10 to the power of
𝑛. So we have a number multiplied by
10 to the power of something.

Well, this is often used for very
big or very small numbers. But there are some conditions that
need to be met. And the key one of those is that 𝑎
is actually greater than or equal to one but less than 10. And it needs to be like that for it
to be in standard form, cause one of the most common mistakes that people make is
actually to have numbers that are actually greater than 10 or less than one, as the
number that we have, multiplied by 10 to the power of something in standard
form.

Okay, so using that, let’s actually
look at our number and see how we could write it in standard form. Well, let’s start off by looking at
what’s our 𝑎 actually going to be when we convert 3040000 into standard form. Well, if we take a look at the
number, then our decimal point will actually have to be after the three, because it
needs to be greater than or equal to one. So it couldn’t be any further to
the left because that would give us 0.3. But it also has to be less than
10. So therefore, we can’t put any
further to the right because that’d be 30. And in that case or anything
greater than that would be greater than 10.

So therefore, we can say that 𝑎 is
gonna be equal to 3.04. And we said 3.04 cause we need to
actually include each part within there. So we need the four because that’s
gonna be the 40000.

Okay, the next stage is to actually
think, right, what are we gonna have to multiply this by to get our original number
of 3040000? Well, we can actually see from what
I’ve demonstrated that we actually would have to multiply by 10 six times, because
if we multiplied it by 10 we get 30.4 and if we multiply it by another 10 we get
304. Multiply it by another 10, we get
3040. Multiply it by another 10, we get
30400. Multiply it by another 10, we get
304000. Multiply it by one more 10, we get
3040000.

So therefore, we can say that our
𝑛 is going to be six, because we’d have to multiply it by 10 to the power of six or
by 10 six times. The other way that we can think of
it if we’re trying to find it would be actually to think, right, we’ve got 3.04 and
we need to make 3040000. What do I need to multiply this
by?

Well, if you want to get three to
3000000, you’d actually have to multiply it by one million. So what we’d have is 3.04
multiplied by 1000000. And again, if you look at the
number of zeros with one million, there are actually six of these, which means it’s
10 multiplied by 10 multiplied by 10 multiplied by 10 multiplied by 10 multiplied by
10. So again, it reinforces the fact
that we said that 𝑛 is gonna be equal to six.

So therefore, if we insert our
values for 𝑎 and for 𝑛 into our general form, we can say that 3040000 in standard
form is going to be 3.04 multiplied by 10 to the power of six. Again, that’s 10 to the power of
six because 10 to the power of six would give us 1000000 and 3.04 multiplied by
1000000 would give us 3040000.

So now we’re gonna move on to part
b. And part b says write 3.7
multiplied by 10 to the power of negative three as an ordinary number. Now the key thing here is the fact
that it’s actually 10 to the power of negative three, because all of this actually
tells us is actually we’re gonna divide the number by 10, then divide the number by
10 again, and then divide the number by 10 again, which is the same as dividing the
number by 1000 or dividing the number by 10 to the power of three.

So one way that we can look at this
and try to think about how we’re gonna get our ordinary number is think about it
using we’ve got here place values. So we’ve got three units and then
seven tens. And that’s because we’ve got
3.7.

As we already said, we want to
divide it by 1000. And therefore, what this means is
that, actually, we’re gonna shift our number, so our digits, three spaces to the
right. So we’re actually gonna move them
down three columns to the right. And when we actually do that, what
we also do is fill in the spaces that were left with zeros.

So when we move them three spaces
to the right, what we’re gonna get is 0.0037. And that’s because, as we said, if
you divide by 1000, it’s three spaces to the right. If we divided by 100, it would be
two spaces to the right. And also our power would be
negative two. So therefore, we can say that 3.7
multiplied by 10 to the power of negative three as an ordinary number is equal to
0.0037.