Rational Exponents - Concept
Any radical can also be expressed as a rational exponent. For example, a cube root is equivalent to an exponent of 1/3; a fourth root is an exponent of 1/4. When using this method to simplify roots, we need to remember that raising a power to a power multiplies the exponents. This topic is important when finding derivatives and in integral calculus.
So up until now we have talked mostly about using exponents that are whole numbers okay? Integers we talked about positive and negative but what we're going to talk about now is exponents that are fractions okay? And don't be scared they behave just the same as the everything else but it's going to draw some comparisons as the things we already know.
Okay, so I have upon the board I have a couple of problems we have 5 squared to the fourth. We know that when we take a power to power we multiply our exponents so this just becomes 5 to the eighth. Okay, square root of 7 squared. When we have a square root with square they cancel out and this just leaves us with whatever is in the inside which is 7. Okay another example this one is incorporating that one half I don't know if you can see is a little bit small but this is just 7 to the one half squared. Okay, when we take a power to a power just like we did up here we multiply our exponents so this just turns into 7 to the one half times 2, one half times 2 is the first so this is just 7 to the first which is 7 as well okay so what does that tell us we took something and squared it we get 7 and we took something and we squared it we got 7, so what that tells us is they're either equal on opposite like 4 and -4 or they're are the same thing and there's no negative signs here so what this really tells us is we have the square root of 7 is equal to 7 to the one half okay what we can do with this is actually make some generalizations okay? We have the nth root of b okay? What we know is that over here, it was the square root and remember this little invisible 2 in front of the square root that is our root okay and that same number actually ended up in the denominator over here so what that tells us about over this is that this is the same thing as b to the 1 over n okay the route is what goes into the denominator of a fractional exponent. Same idea here, this is then a to a to the m and the nth root is just a 1 over n power to power we multiply this turns into a to the m over n and basically the way that I remember this okay is power over root okay our power is m in this case, our root is n so we end up with m over n power over root so dealing with fractional exponents okay and how they tie together with square roots.