Exponential equation with rational answer

so right here we've got three to the a power or eighth power I guess is I don't want to confuse it with the number eight three to the a power is equal to the fifth root of three squared and what we need to figure out is what would a be equal to let's solve for a and I encourage you right now to pause this video and try it on your own well if you have a fifth root right over here one thing that you might be tempted to do to undo the fifth root is to raise it to the fifth power and of course we can't just raise one side of an equation to the fifth power whatever we do to one side we have to do the other side if we want this to still be equal so let's raise both sides of this equation to the fifth power raise both sides to the fifth power now this left hand side we just have to remember a little bit of our exponent properties 3a to the fifth power and if we want to just remind ourselves where that comes from that's the same thing as 3 to the a times 3 to the a times 3 to the a times 3 to the a times 3 to the a well what's that going to be equal to that's going to be 3 to the a plus a plus a plus a plus a power which is the same thing as 3 to the 5a power so the exponent property here is if you raise a base to some exponent and then raise that whole thing to another exponent that's the equivalent of raising the base to an exponent that is the product of these two exponents so we could rewrite this left hand side as 3 to the 5a power 3 to the 5a power is going to be equal to well if you take something that's a fifth root and you raise it to the fifth power then you're just left with what you had under the radical that's going to be equal to 3 squared so now things become a lot clearer 3 to the 5a needs to be equal to 3 squared or another way of thinking about we have the same base on both sides so this exponent needs to be equal to this exponent right over there or we could write that 5 times a needs to be equal to 2 and of course now we can just divide both sides by 5 and we get a is equal to 2/5 2 over 2 over 5 and this is an interesting result and what's neat about this example it kind of shows you the motivation for how we define rational exponents so let's just put this back in into the original expression we've just said or we've just solved for a and we've gotten the 3 to the 2/5 power to the 2 and actually let me color code it a little bit because I think that'll be interesting 3 to the 2 over 5 power 3 to the 2 over 5 power is equal to is equal to the 5th root notice the 5th root so the denominator here that's the root so the 5th root of 3 of 3 squared 3 squared so if you take this base 3 you square it but then you take the 5th root of that that's the same thing as raising it to the 2/5 power notice take this 3 take it to the 2nd power and then you find the fifth root of it or we could if you use this property that we just saw right over here you could rewrite you could rewrite this this is the same thing this is the same thing as 3 squared 3 squared and then you raise that and then you raise that to the 1/5 power we saw that property at play over here you could just multiply these two exponents you get 3 to the 2/5 power and that's the same thing as 3 squared and then find the fifth root of it 3 squared and then you're essentially finding the fifth root of it