University of Michigan
Runs his own tutoring company
Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!
A common mistake that students make is in the rules of exponents. What I have behind me are two different problems with exponents, one where we are multiplying their bases and the other one where we’re taking something to a power. We’re used to dealing with this in sort of you know, numbers, x², x³, or whatever it s, but it’s going to work for variables and hopefully you get a better sense of how this all pieces together.
If we had say x times x to the third, we have one x here, we have 3x here so when you multiply them out together we would end up with x to the 4th. The other way we can do that s we can add our exponents. Here is a 1, here’s a 3, 1 plus 3 is 4. For this case it doesn’t work out quite as nicely because we’re dealing with variables but what’s still going to happen is when multiplying your bases we end up adding our exponents. So what we’re going to do in this case is just have x to the n plus n plus 3. N plus n we can simplify and so this turns out to be x to the 2n plus 3. When we are multiplying our bases we add our exponents.
The next case that we can look at is a exponent to an exponent. Going back to another applied case, this is going to be say x² to the third. What this works out to be is really x² times x² times x². Then we can use the rules from up here and add all these together so 2 plus 2 plus 2 is just going to be 6. What ends up happening is we multiply our 2 exponents. If we have an exponent to an exponent we just multiply and end up with that result.
Here I’m dealing with variables but we still have a power to power so I still want to multiply. What I end up with is x the m times m plus 1. Distributing that m through we end up with x to the m² plus m.
Two different rules for exponents. When you’re multiplying your bases you add, when you have a power to power you multiply. Very commonly messed up, people want to do weird thing whenever you see this kind of thing. Bases same, you're multiplying, add your exponents, power to power, multiply.
