Powers of fractions

let's go through more exponent examples so to warm up let's think about taking a fraction to some power so let's say I have two thirds and I want to raise it to the third power here and we've already learned there's two ways of thinking about this one ways to say let's take three two-thirds so that's one two third to two thirds and three three two-thirds so that's one two three two-thirds and then we multiply them and then we multiply them and we will get and we will get let's see the numerator will be 2 times 2 times 2 which is 8 and the denominator is going to be 3 times 3 times 3 which is equal to 27 now the other way of viewing this is you start with a 1 and you multiply it by 2/3 three times so you multiply by 2/3 once twice three times you will get the exact same result here so let's do one more example like that so let's say I had 4/9 and I want to square it when I raise something to the second power people often say you're squaring it also raising something to the third power people sometimes say you're cubing it but let's raise 4/9 to the second power let's square it and encourage you to pause the video and work this out yourself well once again you could view this as taking four to four ninths taking to four ninths and multiplying them or you could view this as starting with a 1 and multiplying it by 4 ninths 2 times either way your numerator is going to be 4 times 4 which is 16 and your denominator is going to be 9 times 9 which is equal equal to 81