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CCSS.Math: ,

we've already seen how to think about something like 64 to the 1/3 power we saw that this is the exact same thing as taking the cube root of 64 and because we know that 4 times 4 times 4 or 4 to the third power is equal to 64 if we're looking for the cube root of 64 we're looking for a number that that number times that number times that same number is going to be equal to 64 well we know that number is 4 so this thing right over here is going to be 4 now we're going to think of slightly more complex fractional exponents the one we see here has a 1 in the numerator now we're going to see something different so what I want to do is think about what 64 to the 2/3 to the 2/3 power is and here I'm going to use a property of exponents that we'll study more later on but this property of exponents is the idea that let's say with a simple number if I raise something to the third power and then I were to raise that to say the fourth power this is going to be the same thing as raising it to the 2 to the 3 times 4 power or two to the twelfth power which you could also write as raising it to the fourth power and then the third power all this is saying is if I raise something to a power and then raise that whole thing to a power it's the same thing as multiplying the two exponents this is the same thing as 2 to the 12th so we could use that property here to say well look 2/3 is the same thing as 1/3 times 2 so we could go in the other direction we could say hey look well this is going to be the same thing as 64 to the 1/3 power and then that thing squared notice I'm raising something to a power and then raising that to a power if I were to multiply these two things I would get 64 to the 2/3 power now why did I do this well we already know what 64 to the 1/3 power is we just calculated it that's equal to 4 so we could say that this is equal to and I'll write in that same yellow color this is equal to 4 squared this is equal to 4 squared which is equal to 16 which is equal to 16 so 64 to the 2/3 power is equal to 16 the way I think of it let me find the cube root of 6 four which is four and then let me square it and that is going to get me to sixteen now I'll give you an even hairier problem and I encourage you to try this one on your own before I before I work through it so we're gonna work with eight over twenty seven eight over twenty seven and we're gonna raise this thing to the negative to the negative and I'll try to color code it negative 2 over 2 over 3 power through the negative 2/3 power encourage you to pause this and try this you're on your own well the first thing I do whenever I see a negative exponent is to say well how can I get rid of that negative exponent and I just remind myself well the negative exponent really just says take the reciprocal of this to the positive exponent so this is going to be equal to the reciprocal of this is 27 I'm using a different color let me use that light move color so this is going to be equal to 27 over 8 over 8 I just took the reciprocal of this right over here it's equal to 27 over 8 to the positive 2/3 power to the positive 2/3 power so notice all I did got rid of the exponent and took the reciprocal of the base right over here 8 over 27 is the base negative 2/3 is the exponent now how can we handle this well we've already seen that if I have a new enumerator to some power over denominator some power this is another very powerful exponent property this is going to be the exact same thing this is going to be the exact same thing as raising 27 to the 2/3 power to the 2 over third over 3 power over over 8 to the 2/3 power 8 to the 2/3 power this is another very powerful exponent property notice if I have something divided by something and I'm raising the whole thing to a power I can essentially raise the numerator that power over the denominator raised to that power now let's think about what this is well just like we saw before this is going to be the same thing this is going to be the same thing as 27 to the 1/3 power 27 to the 1/3 power and then that squared and then that squared cuz one-third times two is two thirds so I'm gonna raise 27 to the one-third power and then square whatever that is and that is going to be over all this color coding is making this just which a lot of cars this is going to be over 8 to the 1/3 power 8 to the 1/3 power to the and then that's going to be raised to the second power same thing we're doing in the denominator raised 8 to the 1/3 and then square that so what's this going to be well 27 to the 1/3 power 27 to the 1/3 power is the cube root of 27 it's some number that number times that same number times that same number is going to be equal to 27 well it might jump out at you already that 3 to the third is equal to 27 or that 27 to the 1/3 is equal to 3 so the numerator we're going to be we're gonna end up with 3 squared and then in the denominator we are going to end up with well what's 8 to the 1/3 power well 2 times 2 times 2 is 8 so this is 8 to the 1/3 is 2 and then we are going to and then let me do that same orange color 8 to the 1/3 is 2 and then we're going to square that so this is going to simplify to 3 squared over 2 squared which is just going to be equal to 9 over 4 so if you just break it down step-by-step it actually is not too not too daunting