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.