If 2x-y = 3, what's the value of 9^x / 3^y ?
I saw something similar on the SAT I took and thought this was nice and tricky. It also works exponent rules thoroughly, which is one of the biggest parts of algebra. First, put the two numbers in the fraction into the same base. We do this by turning the 9 into 3^2. Thus, we have 9^x = (3^2)^x, and by exponent rules, we know that this is 3^(2x). Now we have a fraction of 3^(2x)/3^(y). Once again, by exponent rules, we know that this simplifies to 3^(2x-y). And voila! We can substitute! We know from the problem that 2x-y = 3, so the expression they give us is simply 3^(2x-y) = 3^3 = 27. Alternatively, you could have solved 2x-y = 3 for x or for y and then done similar steps to above, only substituting one of the two variables in. The remaining variable would cancel, leaving you with the same answer.
What is the value of e^(pi x i)?
If you type it into your google search bar, you'll find out that it's -1. Isn't that kind of weird though? e is a transcendental number that arises from the convergence of a series that's derivative is itself. i is an imaginary number that doesn't even exist in the real world. pi generally is used in circles and triangles as a way of talking about angles. There doesn't seem to be any good reason that sticking them together in this random way would get such a clean result. The trick to proving this one is to look at the expansions of e^(x) using power series. With some magic, it turns out that imaginary exponents can be expressed in terms of trig functions, which is also due to the trig functions' power series. These power series are closely related to Taylor's Theorem and Taylor Series and these concepts become very vital parts of this course. Not only is this cool, but it's incredibly powerful. Taylor's Theorem is described by many as being the most important theorem in Applied Mathematics, and this is where we meet it! With it, we can approximate things we can't solve for explicitly (vital in the world of computing), and if we carry it on infinitely, we get the power series representations of functions, which have nice applications in Calculus and in theory like shown above.
What's a derivative and why should I care?
Back in Algebra, when they spent all that time teaching you how to find the slope of a line in all the different forms the equation could take, the slope was a measure of how steep the line was. A slope of two, for instance, meant that the line would rise two units for every one unit you went to the right. Turns out, curves have slopes too! And that's all a derivative is -- a measure of how step a curve is at any point. If you were to walk along the curve (picture the curve being a side walk, perhaps), the derivative is just an equation that tells you what direction you would be facing at any given point on your walk. If you know about Calculus, you probably already knew that. So the better question is, what does it matter? Well, in intro Calculus, the derivative is already used for some pretty amazing things. When it hits 0, maximum's and minimums can occur in functions, and if you have some big equation that predicts your profit, or a desirable outcome, you can maximize your earnings quickly. Past that, later on in math Calculus and derivatives becomes integral (ha), and studies such as differential equations can help you do a number of crazy things like predict the spread of disease in a population and figure out when it will become a pandemic or just fizzle out! Important stuff! So all and all, the derivative is just the slope of a curve, but it's a pretty exciting and useful thing to know about.