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# Scientific notation word problem: speed of light

CCSS.Math:

## Video transcript

the speed of light is 3 times 10 to the 8th meters per second so as you can tell light is very fast 3 times 10 to the 8 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 that a little bit so 5 times 10 to the 2nd that's 500 500 seconds you have 60 seconds in a minute so 8 minutes would be 480 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 want to find a distance so this goes straight back to the the standard distance is equal to rate times time so they give us the rate the rate is 3 times 10 to the 8th meters per second so it's 3 times 10 to the 8th meters per second that right there is the rate they give us the time the time is 5 times 10 to the 2nd seconds so times 5 times 10 to the second seconds seconds I'll just use that with an S so how many meters so what is the distance what is the distance and so we can just reassociate these or actually move these around from the commutative and the associative properties of multiplication so this right here is the same thing and actually you can multiply the unit's that's called dimensional analysis and when you multiply the units you kind of treat them like variables you should get the right dimensions for a distance so let's just rearrange these numbers this is equal to 3 times 5 3 times 5 I'm just re commuting or INRI associating these numbers so 3 times in this product we're just multiplying everything 3 times 5 times 10 to the 8th times 10 to the 8th times 10 to the 2nd times 10 to the second and then we're going to have m/s so we could write m/s times seconds times seconds and if you treated these like variables this seconds would calculate cancel out with that seconds right there and you 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 8 times 10 squared we have the same base we're taking the product so we can add the exponents so this is going to be 10 to the 8 plus 2 power or 10 to the 10th 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 greater than or equal to 1 and less than 10 this clearly is not less than 10 so how do we rewrite this so we can write 15 as 1.5 this clearly is greater than 1 and less than 10 and if to get from 1 point 5 to 15 you have to multiply you have to multiply by 10 one way to think about it is 15 is 15 point 0 and so you have a decimal here where if we're going to move the decimal one to the left to get us 1.5 we're essentially dividing by 5 then we also 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 times 10 to the 10th not X to the tenth 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 so 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 a huge distance just just so that you can well it's actually almost it's very hard to visualize but anyway hopefully enjoyed that