First things first: I should define what exactly I mean by “hard.” There are scads of ways something can be difficult, and comparing them doesn’t always make sense. Compare, for example, a Rubik’s cube and running a mile in under five minutes. It’s tough to say which is harder, because a.) they’re both ridiculously hard in the first place and b.) they’re completely different skills, so it depends on who you are and what you’re good at.
SAT math is so straightforward a test that only one skill is tested. There are a few different aspects, so the hardest questions can be hard in different ways.
Knowing your stuff
If you don’t already know what’s on the SAT, this part might come as a relief. The test doesn’t go deep into high-level math. There’s no trigonometry, no precalculus, no statistics—nothing that you get in advanced math classes during your last year or two of high school. The most advanced math on the SAT are functions and parabolas, but even then you’ll see just a couple related questions on your test and you only need to know the bare minimum about how they work.
That said, you really have to know your rules when it comes to algebra, geometry, and number properties. There’s nothing particularly advanced, but there is a lot of detail. If you’re not confident with the first few sets of Pythagorean triples or aren’t sure what a negative exponent means, then you have some brushing up to do. The toughest questions do rely on those details, but that’s not what makes them really tough.
Being experienced (and working quickly)
Even if you do know the exponent rules well, how quickly and confidently can you apply them to a question like this one?
If k, n, x, and y are positive numbers satisfying x^(-4/3) = k^(-2) and y^(4/3) = n^2, what is (xy)^(-2/3) in terms of n and k?
It’s easy to slip up if you’re working with rules that you don’t use often, and that’s precisely what this question relies on to make it difficult. (This is a real SAT question from a past test, by the way.) All of the math that goes into it is relatively straightforward, but if you’re not used to answering questions that work with tricky exponents, then you’re in danger of faceplanting.
This is what makes the SAT special, and it’s why people find themselves actually enjoying SAT questions more than they might expect. The math is part puzzle, really. When a question is really tough, you have to do more than just know the rules and apply them. You also have to figure out the fastest way to arrive at an answer and see unexpected questions between the information.
The hardest SAT math is not based entirely on what you’ve learned or memorized; it wants you to explore and find patterns for yourself. Check out some examples to see what I mean—can you find shortcuts, when they’re possible? Can you remember all the rules you need to and apply them when working under pressure?