We’d express this really small
number in scientific notation. Let’s just see how
small it is. It is one, two, three,
four, five, six, seven, eight, nine zeroes. And then a 3457. All of that is behind
the decimal point. Now this is a bit of review
about scientific notation. It means expressing a number
in a form– let me give an example, it could be 5 times
10 to the fifth power. This would be an example
of writing something in scientific notation. And when you write something
in scientific notation, the number that you’re multiplying
by a power of ten, that number right there has to be greater
than or equal to 1, and it should be less than 10. And of course, this right here
is going to be a power of 10. So for example, if I were to
write 5 times 9 to the fifth, not scientific notation. If I were to write 50 times
10 to the fifth, not quite scientific notation. Sometimes this kind of gets by
for scientific notation while you’re doing computation. But if you really wanted to
write this in scientific notation, you would write this
as 5 times 10, which is 50 times 10 to the fifth. And 10 times 10 to the fifth
is 10 to the sixth. So you’d write this as 5
times 10 to the sixth. So your goal is always for this
number right here to be greater than or equal
to 1, less than 10. Likewise, if I wrote 0.9 times
10 squared, once again, not in scientific notation. You would want to write 0.9 as
9 times 10 to the negative 1 times 10 squared. And then this becomes
9 times 10. Which makes sense, because
0.9 times 100 is 90. 9 times 10 is 100. We could say 9 times 10
to the first power. So that was a bit of review. Let’s do it for this
problem right here. I guess a quick way to think
about converting this to scientific notation, think about
how many leading zeroes there are including the first
non zero number right there. So there are 10 digits to the
right if you include this 3. You have nine zeroes and
then you have this 3. So a very quick way to do it,
and I’ll explain in a second why it works, is we could
rewrite this right here as 3.457 times 10 to the, we had
nine zeroes, and then I count the 3, 10. So times 10 to the negative
10 power. And the reason why this makes
sense is, when you multiply something to a negative
power, you shift the decimal over to the left. Let me write it this way. Start with 3.457. And when you multiply it times
10 to the negative 10, you’re going to shift the decimal
to the left 10 spaces. So if you shift the decimal
to the left 10 spaces, what do you get? So if you shift it once, twice,
three, four, five, six, seven, eight, nine, ten. And of course, you can’t just
leave open space here. There’s going to be zeroes
there when you shift it. So you’re going to have one,
two, three, four, five, six, seven, eight, nine zeroes. And you can put a zero out here
just to make sure that you understand that there is a
zero in front of that decimal. And you get exactly what
we got up there. Another way to think about it
is, if you wanted to start with this really small number
and you wanted to get to 3.457, you could multiply
it by 10 to the tenth. That would shift this decimal
10 spaces to the right. So you could do this. And these are all the multiple
ways of looking at the exact same thing. You could write that 3.457 is
equal to this character. Or I could even copy and
paste this character. Times 10 to the tenth power. If you multiply by a positive
exponent, every time you multiply by 10 it shifts the
decimal to the right. It makes the number bigger. So if you multiply this by 10
to the tenth, this guy is going to move one, two, three,
four, five, six, seven, eight, nine, ten. It would shift it right there. And you’d get 3.457. So that’s the whole idea. 3.457 is this number times
10 to the tenth. So if you wanted to solve for
this number, you would multiply both sides of this
equation by 10 to the negative 10. So times 10 to the negative 10,
or that’s the equivalent of dividing by 10
to the tenth. These cancel out. Negative tenth is 10 to
the zero or just 1. And you get this thing over
here being equal to 3.457 times 10 to the negative 10. I just wrote it in a different
order over here.