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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. Δημιουργήθηκε από Σαλ Καν και Monterey Ινστιτούτο Τεχνολογίας και Εκπαίδευσης.

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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.