So generally speaking, it's going to be really important that you have a really good handle on algebra and algebraic expressions in physics. So the first thing we're going to talk about is how to simplify expressions. For example, what I'm going to show you is that we can take expressions like this, and through subtraction, multiplication, and addition, all those things, we can actually just simplify this to something like x. And I’m going to show you how this works. Basically, we take long expressions, and we make them simpler by reducing the number of terms. We just write them in fewer terms. For example, if I have this expression over here, basically, what I'm going to do is I’m going to distribute anything that's on the outside of parentheses, then I can group together the terms that are similar, and then I can combine them by adding and subtracting. So the way this works is that I have \(2x+3\), don't change anything there, but I have to distribute the 4 into everything that's inside the parenthesis, so I get \(4x + 8\). Now what I do is I just group the things that are like each other, like the \(2x\) and \(4x\) and the \(3\) and the eights. So this is \(2x + 4x\), which just becomes \(6x\). And then I have 38, which just combined down to 11. So this whole entire expression here really just becomes \(6x + 11\), and that is the simplest way I can write that. Alright? Now let's move on.

Another thing you're going to have to be really good at, or just familiar with, is expressions with exponents. So I'm just going to go over really quickly what exponents are. Basically, exponents just represent repeated multiplication. We'll do this a lot when we're talking about scientific notation. So just some basics here. If you have, like, a number like 4 multiplied by itself 5 times, it's really sort of tedious to write out. So what we do is we write this with a sort of shorthand notation 4 with a little superscript of 5. It's a little tiny 5 in the top right corner. The way we say this is it's 4 to the fifth power. The 4 is the base. It's the number or variable that you're multiplying, So that's the 4, and the exponent or the power is how many times you're multiplying that base by. So base is 4 and the power is 5. Generally, the way that we write any exponent is if you have something, it could be a number or a variable a, and if you're multiplying it by itself \(n\) times, then we just write it as \(a^n\). Alright? Now so that's just how generally exponents work.

What you're also going to have to be pretty sort of pretty you know, have a good handle on is how to manipulate exponents. So there's a couple of rules that you're going to probably use, more frequently than others, and I sort of summarize them in a little table here. So there's some important ones like the product and quotient rule. These come up all the time. This is when you have something like \(4^2 \times 4^1\). You're multiplying things of the same base. Basically, what you're going to do here is you're going to add the exponents. So this is like \(4^{2+1}\), which is \(4^3\). So when you're multiplying things that are the same base, you add their exponents. And then when you're dividing things of the same base, like \(4^3\) divided by \(4^1\), you're actually just subtracting the exponents. This becomes \(4^2\). So when you're dividing, then you subtract the exponents. A couple of other really quick ones here, anything to the 0 power always just equals 1. That's just a rule in math that you should know. Whenever you have negative exponents like \(4^{-2}\), basically, the way this works is you're going to sort of flip it to the bottom of a fraction. So now this becomes \(1/4^2\) and you drop the minus sign. So generally, what happens is when you have negative exponents on the numerator, you flip them to the bottom, the denominator, write them with a positive exponent. But you also could have something like a negative exponent in the denominator. In that case, you flip it to the top and you write with a positive exponent. So you always do the reciprocal and then rewrite it with a positive exponent. Alright.

Another one's called the power rule, and this is basically where you have a power that's on top of another power on the outside of a parenthesis. Basically, what you're going to do here is you're going to multiply their exponents. So it's kind of similar to what we do with product and quotient, but now you're multiplying the exponents, so this becomes \(4^6\). So you multiply them. And then finally, power of a product. This basically just means if you have something like \(3 \times 4^2\), you distribute the exponent to everything that's on the inside. So this becomes \(3^2 \times 4^2\). Same things happens with quotients. You basically just distribute the exponent to everything that's inside the parenthesis. This becomes \(12^2/4^2\). So really, what you're doing here is you're just distributing the exponent to every term that's in the parentheses. And in the case where you have a fraction, you just distribute it to the numerator and denominator. Alright.