Multiplying in Scientific Notation

We're asked to
multiply 1.45 times 10

to the eighth times 9.2 times
10 to the negative 12th times

3.01 times 10 to
the negative fifth

and express the product in
both decimal and scientific

notation.

So this is 1.45 times 10
to the eighth power times--

and I could just write the
parentheses again like this,

but I'm just going to write
it as another multiplication--

times 9.2 times 10
to the negative 12th

and then times 3.01 times
10 to the negative fifth.

All this meant, when I wrote
these parentheses times next

to each other, I'm
just going to multiply

this expression
times this expression

times this expression.

And since everything is
involved multiplication,

it actually doesn't matter
what order I multiply in.

And so with that in mind,
I can swap the order here.

This is going to be the
same thing as 1.45-- that's

that right there-- times
9.2 times 3.01 times

10 to the eighth--
let me do that

in that purple color-- times
10 to the eighth times 10

to the negative 12th power times
10 to the negative fifth power.

And this is useful
because now I have

all of my powers of
10 right over here.

I could put parentheses
around that.

And I have all my non-powers
of 10 right over there.

And so I can simplify it.

If I have the same base
10 right over here,

so I can add the exponents.

This is going to be 10 to
the 8 minus 12 minus 5 power.

And then all of this
on the left-hand side--

let me get a calculator
out-- I have 1.45.

You could do it by hand, but
this is a little bit faster

and less likely to make a
careless mistake-- times 9.2

times 3.01, which
is equal to 40.1534.

So this is equal to 40.1534.

And of course, this is going
to be multiplied times 10

to this thing.

And so if we simplify
this exponent,

you get 40.1534 times
10 to the 8 minus 12

is negative 4, minus
5 is negative 9.

10 to the negative 9 power.

Now you might be tempted
to say that this is already

in scientific notation because
I have some number here

times some power of 10.

But this is not quite
official scientific notation.

And that's because
in order for it

to be in scientific notation,
this number right over here

has to be greater than or
equal to 1 and less than 10.

And this is, obviously,
not less than 10.

Essentially, for it to be
in scientific notation,

you want a non-zero
digit right over here.

And then you want
your decimal and then

the rest of everything else.

So here-- and you want
a non-zero single digit

over here.

Here we obviously
have two digits.

This is larger than 10-- or this
is greater than or equal to 10.

You want this thing
to be less than 10

and greater than or equal to 1.

So the best way to do that
is to write this thing

right over here in
scientific notation.

This is the same thing
as 4.01534 times 10.

And one way to think about
it is to go from 40 to 4,

we have to move this
decimal over to the left.

Moving a decimal over to
the left to go from 40 to 4

you're dividing by 10.

So you have to multiply by
10 so it all equals out.

Divide by 10 and
then multiply by 10.

Or another way to write it, or
another way to think about it,

is 4.0 and all this stuff times
10 is going to be 40.1534.

And so you're going to have
4-- all of this times 10

to the first power, that's
the same thing as 10-- times

this thing-- times 10 to
the negative ninth power.

And then once
again, powers of 10,

so it's 10 to the first
times 10 to the negative 9

is going to be 10 to the
negative eighth power.

And we still have this 4.01534
times 10 to the negative 8.

And now we have written
it in scientific notation.

Now, they wanted
us to express it

in both decimal and
scientific notation.

And when they're asking us to
write it in decimal notation,

they essentially want us to
multiply this out, expand this

out.

And so the way to think about
it-- write these digits out.

So I have 4, 0, 1, 5, 3, 4.

And if I'm just
looking at this number,

I start with the
decimal right over here.

Now, every time I divide by
10, or if I multiply by 10

to the negative 1, I'm moving
this over to the left one spot.

So 10 to the negative
1-- if I multiply by 10

to the negative 1, that's the
same thing as dividing by 10.

And so I'm moving the
decimal over to the left one.

Here I'm multiplying by
10 to the negative 8.

Or you could say I'm dividing
by 10 to the eighth power.

So I'm going to want to move
the decimal to the left eight

times.

And one way to
remember it-- look,

this is a very, very,
very, very small number.

If I multiply this, I
should get a smaller number.

So I should be moving
the decimal to the left.

If this was a
positive 8, then this

would be a very large number.

And so if I multiply
by a large power of 10,

I'm going to be moving
the decimal to the right.

So this whole thing
should evaluate

to being smaller than 4.01534.

So I move the decimal
eight times to the left.

I move it one time to the left
to get it right over here.

And then the next seven times,
I'm just going to add 0's.

So one, two, three, four,
five, six, seven 0's.

And I'll put a 0 in front of
the decimal just to clarify it.

So now I notice, if you include
this digit right over here,

I have a total of eight digits.

I have seven 0's, and
this digit gives us eight.

So again, one, two, three,
four, five, six, seven, eight.

The best way to
think about it is,

I started with the
decimal right here.

I moved once, twice, three,
four, five, six, seven,

eight times.

That's what multiplying times
10 to the negative 8 did for us.

And I get this number
right over here.

And when you see a
number like this,

you start to appreciate
why we rewrite things

in scientific notation.

This is much easier to-- it
takes less space to write

and you immediately know
roughly how big this number is.

This is much harder to write.

You might even
forget a 0 when you

write it or you might add a 0.

And now the person has to sit
and count the 0's to figure out

essentially how large--or get
a rough sense of how large this

thing is.

It's one, two, three, four,
five, six, seven 0's, and you

have this digit right here.

That's what gets
us to that eight.

But this is a much, much more
complicated-looking number

than the one in
scientific notation.