Simplify the expression with NO negative exponents. 2 − 1 ⋅ 2 4 2^{-1}\cdot2^4

Coming off the practice problem that we just did before, we had two to the negative one plus two to the fourth power, but now this is going to be different because we're going to multiply these terms. We're still going to simplify with no negative exponents. Let's get started here. Now, again, typically we deal with negative exponents before we do things like multiplication. However, there's one important difference here, which is that we actually have the same base. We have two raised to two different exponents, so you could just do the product rule to sort of shortcut some of this, but I'm not going to do that. Let's just actually go ahead and work this out properly. We've got two to the negative one power times two to the fourth. So, I should rewrite this as, because I have a negative exponent here, I need to rewrite this with a positive exponent by writing the reciprocal. This is two to the negative one power over one, so one way I can rewrite this is one over two to the positive one power. Now, I'm going to multiply and this is just two to the fourth power. There's no negative exponent here, so I don't have to rewrite this. So, this is one over two times two to the fourth power. Now, I can simplify this because I can kind of imagine that this two to the fourth power is really just two to the fourth power over one because of multiplying fractions, remember, you always just multiply straight across, right? So, this situation over here, this expression becomes two to the fourth power divided by two. Now, what happens here is that two really is just two to the one power, and so we can see here is we have the same exact base that has two different exponents. So, we can use the quotient rule for this. So, remember the quotient rule just said that whenever you have the same base raised to two different exponents, you subtract their exponents. This is the same thing as two to the four minus one power, and this just becomes two to the third power. This would be sort of the more proper full way to do this by dealing with the negative exponents and then using your other exponent rules. However, one of the things I just want to offer you as a shortcut that you can deal with later on problems is that if you already notice off the bat that your two bases are the same, for example, we have two to the negative one times two to the fourth, then one of the things you can just directly go and do is you can go ahead and just add their exponents together. This is the same thing as two to the negative one plus four, right? That's the product rule and what that actually gets us is two to the negative one plus four is just two to the third power. So, again, this is just a sort of quicker way to get to the same exact answer with less steps but still following your exponent rules. Either way, just go ahead and know that this is the correct answer. Let me know if you have any questions. Thanks for watching.