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# Using exponents with powers of 10

CCSS.Math:

## Video transcript

so let's think about what happens when we multiply by ten and see if we can see some type of a pattern so let's just start with one and multiply by multiply that 1 by 10 and this of course is equal to 10 well you knew that already but let's multiply it by 10 again so let's do 1 times 10 and now let's multiply by another 10 let's multiply by another 10 so what's this going to be this is going to be 1 times 10 which is 10 and then that 10 times 10 so it's literally 10 tens or 100 now let's multiply by 10 again so we're going to have 1 times 10 times 10 times 10 times times 10 what is this going to be equal to well we already know that 1 times 10 times 10 is 100 and then we're going to multiply that times 10 again so we're going to literally have ten 100's which we know is the exact same thing as 1,000 so what's the pattern here well here we multiplied by 10 once and we have one zero here we multiplied by 10 twice and we have two zeros here we multiplied by 10 three times and we have three zeros in our answer let's do it again although it looks like a pretty clear pattern let's multiply by ten four times so we'll start with the one well multiply by 1/10 that's our first 10 then our second 10 then our third 10 and now our fourth 10 well what's this going to be equal to well it's going to be this thing this thing times 10 tiya or this thing I should say times 10 or a thousand times 10 which is 10 thousands so let me write that 10 10,000 so once again we started with a 1 we multiplied by 10 4 times and we got 10,000 one followed by one two three four zeros let me make that clear we start with the one we multiplied by 10 one two three four times and our product is one followed by one two three four zeros so I think you see the pattern here but there might be something you're craving you're like hey Sal this is a pain to do this repeated and multiplication I wish there was some type of operation that just did it for us the same way that back in the day when you were doing repeated addition where if you were saying ten plus ten plus ten plus ten you said hey this is the same thing as four times ten there should be another thing for repeated multiplication if I have one times 10 times 10 times 10 times 10 which is really the same thing as taking four tens and multiplying them together because the 1 doesn't change the value so 10 times 10 times 10 times 10 you're saying hey there should be some type of shorthand for this and you're right there is some type of shorthand for this and it's called an exponent so the same way if you take 4 tens and add them up that's the same thing as 4 times 10 if you take 4 tens and take their product this is the same thing as 10 to the fourth power 10 to the fourth power literally takes means taking four tens and multiplying them together or you could view it as starting with a 1 and multiplying by 10 4 times so this is also equal to 10 to the fourth power this right over here we started with 1 we multiplied by 10 3 times which is the same thing as taking three tens and multiplying them together this is 10 to the third power this right over here is 10 squared or 10 to the second power and this right over here we just have 1 10 right over here we just started with the one we multiplied by 10 once this is 10 to the first power and just so you know a little bit of the terminology here 10 is our base the base is 10 and this for this for that we have right over here this is the exponent the exponent so given that notation let's just do one practice let's figure out let's the about what I'll do new colors what 10 to the 6th power is and I encourage you to pause the video and think about it well there's two ways to think about it you could view this as taking tes six tens so 10 10 10 10 10 and 10 and multiplying them together so we multiply them all together or you could view it as starting with a 1 and multiplying it by 10 6 times but either way what are you going to get well if you're multiplying one by 10 6 times you're going to end up with one followed by six zeros we already saw this pattern so it's going to be 1 followed by 1 2 3 4 5 6 zeros or 1 million so 10 to the 6th power is the same thing as 1 being multiplied by 10 6 times or taking 6 tens and multiplying them together and that's going to be equal to one followed by six zeros which is this pattern we saw or 1 million