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# Scientific notation word problem: speed of light

Amazingly, we can figure out how far the sun is from the earth if we know how to multiply numbers in scientific notation. Created by Sal Khan and Monterey Institute for Technology and Education.

Video transcript

The speed of light is 3
times 10 to the eighth meters per second. So as you can tell,
light is very fast, 3 times 10 to the eighth
meters per second. If it takes 5 times 10 to the
second power seconds for light to travel from the
sun to the earth-- let's think about
that a little bit. 5 times 10 to the second,
that's 500 seconds. You have 60 seconds in
a minute, so 8 minutes would be 480 seconds. So 500 seconds would be
about 8 minutes, 20 seconds. It takes 8 minutes,
20 seconds for light to travel from the
sun to the earth. What is the distance, in meters,
between the sun and the earth? They're giving us a rate. They're giving us a speed. They're giving us a time. And they want to
find a distance. This goes straight back
to the standard distance is equal to rate times time. So they give us the rate. The rate is 3 times 10 to
the eighth meters per second. That right there is the rate. They give us the time. The time is 5 times 10
to the second seconds. I'll just use that with a S. How many meters? So what is the distance? And so we can just move these
around from the commutative and the associative
properties of multiplication. And actually, you can
multiply the units. That's called
dimensional analysis. When you multiply the units,
you kind of treat them like variables. You should get the right
dimensions for distance. So let's just rearrange
these numbers. This is equal to 3
times 5-- I'm just commuting and reassociating
these numbers and this product, because we're just multiplying
everything-- 3 times 5 times 10 to the eighth times
10 to the second. And then we're going to
have meters per second times seconds. And if you treated
these like variables, these seconds would cancel out
with that seconds right there, and you would just be left with
the unit meters, which is good, because we want a
distance in just meters. How does this simplify? This gives us 3 times 5 is 15. 15 times 10 to the
eighth times 10 squared. We have the same base. We're taking the product,
so we can add the exponents. This is going to be 10 to
the 8 plus 2 power, or 10 to the 10th power. Now you might be tempted
to say that we're done, that we have this in
scientific notation. But remember, in
scientific notation this number here has
to be greater than or equal to 1 and less than 10. This clearly is
not less than 10. So how do we rewrite this? We can write 15 as 1.5. This clearly is greater
than 1 and less than 10. And to get from 1.5 to 15,
you have to multiply by 10. One way to think about
it is 15 is 15.0, and so you have a decimal here. If we're moving the decimal
one to the left to make it 1.5, that's essentially
dividing by 10. Moving the decimal to the left
means you're dividing by 10. If we don't want to change
the value of the number, we need to divide by 10
and then multiply by 10. So this and that
are the same number. Now 15 is 1.5 times
10, and then we have to multiply that
times 10 to the 10th power, this right over here. 10 is really just 10
to the first power. So we can just
add the exponents. Same base, taking the product. This is equal to 1.5 times
10 to the 1 plus 10 power, or 10 to the 11th power. And we are done. This is a huge distance. It's very hard to visualize. But anyway, hopefully
you enjoyed that.