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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. So just let's think about a little bit. So 5 times 10 to the second, that's 500. 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. So it takes 8 minutes 20 seconds for light
to travel from the sun to the earth. What is the distance in meters between,
between the sun and the earth? So they're giving us a rate, they're
giving us a speed, they're giving us a time. And they wanna find a distance. So this goes straight back to the, the standard to 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. So it's 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. So times, 5 times 10 to the second seconds, seconds, I'll just use that with
an s. So how many meters? What is the distance? What is the distance? And so we can just re-associate these or
actually move this around from the commutative and
dissociative properties of multiplication. So this right here is the same thing. And actually you can multiply the units. That's called dimension analysis. And when you multiply the units you kind
of treat them like variables, you should get the
right dimensions for distance. Let's just rearrange these numbers. This is equal to 3 times 5, 3 times 5. I'm just re-commuting or re-associating
these numbers. So, three times, in this product, so we're
just multiplying everything. 3 times 5 times 10 to the eighth, times 10
to the eighth, times 10 to the second, times
10 to the second. And then we're going to have meters per second, so we could write meters per
second. Time seconds, times seconds. And if you treated these like variables
these seconds would calc, would cancel out with that seconds
right there and you'll just be left with the unit meters,
which is good because we want a distance in meters,
in just meters. So 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 exponent. So this is gonna be 10 to the eighth plus
2 power or 10 to the tenth power. Now, you might, 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 less greater than or equal to one and less
than ten. This clearly is not less than ten, so how
do we rewrite this. We can write fifteen as 1.5. This clearly is greater than 1 and less
than 10, and to get from 1.5 to 15, we have to multiply, you have to multiply by 10, one
way to think about it is 15 is 15.0, so you have a decimal here, Where, if we're gonna move
the decimal one to the left to get us 1.5, we're
essentially dividing by 5, then we also sorry, 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 wanna change the value of the
number, we either divide by 10. And then multiply it 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, times 10 to the tenth, not X to the 10th, times 10 to
the tenth power, this right over here. 10 is really just 10 to the first power,
so we can just add the exponent, same base taking the product, so this is equal to
1.5 times 10 to the 1 plus 10 power, or 10 To the eleventh
power, and we are done. This is a huge, a huge distance, this just
so that you can well, it's actually almost, it's
very hard to visualize. But anyway, hopefully you enjoyed that.