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### Μάθημα: 8η τάξη > Ενότητα 1

Μάθημα 10: Arithmetic with numbers in scientific notation- Πολλαπλασιασμός & διαίρεση με επιστημονικό συμβολισμό
- Multiplying three numbers in scientific notation
- Πολλαπλασιασμός & διαίρεση με επιστημονικό συμβολισμό
- Subtracting in scientific notation
- Adding & subtracting in scientific notation
- Simplifying in scientific notation challenge

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# Simplifying in scientific notation challenge

Sal simplifies a very hairy expression in scientific notation. Δημιουργήθηκε από τον Σαλ Καν.

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Let's see if we can simplify
this expression right over here and write it in
scientific notation. So the first thing
I want to do, well I already have parts
of this expression that are written in
scientific notation. For my brain, to simplify
the multiplication, I like to write everything
into scientific notation and then do whatever
I need to do to get the final product
into scientific notation. So this part right
over here, 0.2, is not in scientific notation. In order for it to be
scientific notation, it would have to be some
number between 1 and 10, not including 10-- so
greater than or equal to 1, less than 10-- being
multiplied by some power of 10. And this is clearly less than 1. But we could just
view this as, look, this is in the tenths place. This is 2 times 1/10. 1/10 is 10 to the negative 1. So this is the same
thing as 2 times 10 to the negative 1-- same
thing as 2 times 1/10. Now if we look in
the denominator, in blue, we have this part. It is written in
scientific notation. But this green part is not. But we could easily write it
as-- this is five 10,000's. 10,000 is 10 to the fourth. So this is the same thing as 5
times 10 to the fourth power. And you see that it has
one, two, three, four 0's. So now let's take the
product in the numerator and the denominator. So in the numerator,
I'm just going to swap the order in
which I'm multiplying. I'm just multiplying
a bunch of stuff. 4.6 times 10 to the sixth times
2 times 10 to the negative 1. It doesn't matter what
order I multiply them in. So I could rewrite this
as 4.6 times 2 times 2 times 10 to the sixth-- I'm
switching colors-- times 10 to the negative 1. And then in the denominator,
let me just write the 5 times 2.3 times 10 to the fourth
times 10 to the negative 2. And now let us attempt
to simplify this thing. So here, we have 4.6 times 2. Let me circle that. So 4.6 times 2 is 9.2. So that's 9.2. And then 10 to the sixth
times 10 to the negative 1-- we have the same base. We're taking the product. We can add the
exponents-- is going to be 10 to the 6 minus 1
or 10 to the fifth power. So we've simplified
our numerator. And now in our
denominator, let's see. 5 times 2.3-- 5 times 2 is 10. 5 times 0.3 is 1.5, so
it's going to be 11.5. So this is going to be 11.5. And then if I multiply
10 to the fourth times 10 to the negative 2, that's going
to be 10 to the 4 minus 2 or 10 squared-- times 10
to the second power. And now I can divide
these two things. So this is going to
be equal to-- we'll have to think about
what 9.2 over 11.5 is. But actually let me
just do that right now, get a little practice
dividing decimals. Let me get some
real estate here. Let me do that in
the same color. 9.2 divided by 11.5--
well if we multiply both of these times 10, that's
the exact same thing as 92 divided by 115. We're essentially
moved the decimal to the right for both of them. And let me add some zeros
here because I suspect that I'm going to
get a decimal here. So let's think what
this is going to be. Let's think about this. Well 115 doesn't go into 9. It doesn't go into 92. It does go into 920. And I'm going to eyeball and
say that it will go eight times. Let's see if that works out. So I have my decimal here. That's a 0. 8 times-- 8 times 5 is 40. 8 times 11 is 88. And then 88 plus 4 is 92. Oh, it went in
exactly, very good. So 920, we have no remainder. So 9.2 divided by 11.5
simplified to 0.80. And then 10 to the fifth
divided by 10 to the second, we have the same base,
and we're dividing. So we can subtract
the exponents. That's going to be
10 to the 5 minus 2. So this right over
here is going to be 10 to the third power-- so
times 10 to the third power. Now, are we done? Well in order to be done,
this number right over here needs to be greater than or
equal to 1 and less than 10. It is clearly not greater
than or equal to 1. So how can we rewrite this
as the product of something that is greater than or
equal to 1 and less than 10 and some power of 10? Well this 8 right over here,
this is in the tenths place. It's 8/10, 8 times 1/10. So this is going to be the
same thing as 8 times 10 to the negative 1 power. And then we have this 10 to
the third here-- so times 10 to the third power. We'll do that in
that other color. And now we have the same base. Just add the exponents. So this is going to
be equal to 8 times 10 to the 3 minus 1-- so
8 times 10 squared. And we're done. We've simplified our
original expression.