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Video transcript

we already know a good bit about about exponents for example we know if we took the number 4 and raise it to the third power this is equivalent to taking three fours and multiplying them or you could also view it as starting with a 1 and then multiplying the 1 by 4 or multiplying that by 4 3 times but either way this is going to result in 4 times 4 is 16 times 4 is 64 we also know a little bit about negative exponents so for example if I were to take 4 to the negative 3 power we know this negative text tells us to take the reciprocal 1 over 4 to the 3rd and we already know 4 to the 3rd is 64 so this is going to be one sixty-fourth now let's think about fractional exponents so we're going to think about what is 4 to the 1/2 power I encourage you to pause the video at least take a guess about what you think this is so the mathematical convention here the mathematical definition that most people use or that frankly all people use here is that 4 to the 1/2 power is the exact same thing as the square root of 4 and we'll talk in the future about why this isn't and the reason why this is defined this way is it has all sorts of neat and elegant properties when you start when you start manipulating the actual exponents but what is the square root of 4 that especially the principal root mean well that means well what is a number that if I were to multiply it by itself or if I were to have two of those numbers and I were to multiply them times each other that same number I'm going to get 4 well what times what what what time's itself is equal to 4 well that's of course equal to 2 and just to get a sense of why this is this starts to work out well remember we could have also written that 4 is equal to is equal to 2 squared so you're starting to see something interesting 4 to the 1/2 is equal to 2 2 squared is equal to 4 so let's get a couple more examples of this just so you make sure you you you get what's going on I encourage you to pause it as much as necessary and try to figure it out yourself so based on what I just told you what do you think 9 to the 1/2 power is going to be well that's just the square root of 9 the principle root of 9 that's just going to be equal to 3 and likewise we could have also said that 3 squared is or let me write it this way that 9 9 is equal to 3 squared these are both true these are both true statements let's do one more like this what is 25 to the 1/2 I'm going to be well this is just going to be 5 5 times 5 is 25 or you could say that 25 is equal to 5 squared now let's think about what happens we take something to the 1/3 power so let's imagine taking 8 to the 1/3 power so the definition here is that taking something to the 1/3 power is the same thing as taking the cube root of is the same thing as taking the cube root of that number and the cube root is saying well what number if I had three of that number and I multiply them that I'm going to get eight so something times something times something is eight well we already know that 8 is equal to 2 to the third power so the cube root of 8 or 8 to the one-third is just going to be equal to 2 this says hey give me the number that if I take if I say that number times that number times that number I'm going to get 8 well that number is 2 because 2 to the third power is 8 do a few more examples of that what is 64 to the 1/3 power well we already know that 4 times 4 times 4 is 64 so this is going to be this is going to be 4 and we already wrote over here that 64 is the same thing as 4 the third I think you're starting to see a little bit of a pattern here a little bit of symmetry here and we can extend this idea to arbitrary arbitrary rational exponents so what's happening what happens if I were to raise let's say I had let me think of a good number here so let's say at 32 I have the number 32 and I raise it to the 1/5 power so this says hey give me the number that I heard a multi why that number or I would repeatedly multiply that number five times what is that I would get 32 well 32 is the same thing as 2 times 2 times 2 times 2 times 2 so 2 is that number that if I were to multiply it 5 times then I'm going to get 32 so this right over here is 2 or another way of saying this kind of same statement about the world is that 32 is equal to 2 to the fifth power