Great question, and I admire your commitment to thoroughly understanding these concepts!
One suggestion - try to get as much mileage as you can out of each question:
-Ask yourself how they could ask it slightly differently to make it a bit tougher. By doing this, you'll develop a deeper understanding of the concepts themselves, and learn to anticipate ways that the test will tweak familiar themes to make them more difficult.
-When you get to the end of a problem and don't fully understand why the rule worked, even if you got it right, see if you can prove the rule to yourself. We don't do that enough in life - just ask yourself "why?" when you perform a step that you know you're allowed to do but haven't ever thought about, and you can better understand the fundamentals behind it. When I took the test, I just about blanked on the rule for solving (x^y)^z - I was pretty sure you multiplied y and z, but couldn't completely talk myself out of the thought that, maybe, you're supposed to add them. Quickly, I proved it to myself using small numbers: (2^2)^3 would be 4^3, or 64. 64 = 2^6, and not 2^5, so I knew I had to multiply, not add. After the test, I broke it down further - x^2 is the same as x*x. And y^3 is y*y*y. So, (x^2)^3 is (xx)^3, or xx*xx*xx. It's six values of x multiplied together. If you prove some of these rules to yourself, you'll never forget them, because you know exactly why the rule is the way it is.
You may also want to note any concepts that you know haven't "clicked" yet and search for drills that are specific to those concepts. Purple Math (
www.purplemath.com) has a lot of these for algebra concepts, and there are quite a few study resources online for K-12 students that will give you specific drills on, say, exponential equations, or coordinate geometry, or whatever the skills are. If you're not quite there on a concept or two, it's probably worth your while to spend some time on the concept itself before burning through GMAT problems. The Veritas Prep "Math Essentials" lesson gives you quite a bit of that kind of work, but you can always search for a little more.
Brian Galvin
GMAT Instructor
Chief Academic Officer
Veritas Prep
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