Powers of 2

From my early days as an undergrad math student, I knew that powers of 2, the numbers of the form 2ⁿ, have many properties, and are very important in many areas of mathematics.
However, it was only after I became a math tutor that I realized how common questions involving powers of 2 are in standardized tests.

Before becoming a math tutor I had held positions at colleges and high schools as an associate professor, adjunct professor, math teacher, and teaching assistant. Up to that point I had memorized only the first six natural powers of 2, from 2¹ to 2⁶, these are:
2, 4, 8, 16, 32, and 64, the powers of 2 that are two-digit numbers.

Later, when I became a math tutor, and I realized how often standardized tests present students with questions involving powers of 2, then I memorized larger powers of 2, up to 2¹², or:
128, 256, 512, 1024, 2048, and 4096.

It is really a big advantage at test time, for a student to be able to recognize a power of 2 when they see such a number in a problem, because then they can write that number in its exponential format, and use the algebraic rules of exponents to work out the problem’s math easier and faster than dealing with the number in its decimal form, using only arithmetic operations, because the longer route is more time-consuming, and more attention-intensive, therefore riskier and less efficient.

So, if you are preparing for a standardized test, from the CBEST or the ASVAB to the GMAT or the CSET, know your powers of 2. Memorize them well from 2¹=2 to at least 2¹⁰=1024, and you’ll increase your chances of scoring a few extra points in the exam.

Ratio Word Problems in Standardized Tests

Look for all the numbers the problem does not show you

Standardized tests like GMAT, GRE, CBEST, and ASVAB include ratio word problems. These may be, for example, problems about mixing water with alcohol, or about the ratio of girls to boys in a classroom, or any other type of situation where it makes sense to talk about ratios. There is a consistent pattern that shows in nearly every ratio word problem found in standardized tests. They give you the basic proportion between two parts, and then they ask you a question about the total. Or they give you the ratio between the total and one of the parts, and then they ask you a question about the other part. To give a simple example, let’s consider this problem:
In a certain school, the ratio of girls to boys is 5 to 7. How many students are there in a classroom with 15 boys, if the same girls to boys ratio applies to that classroom?
Notice how the given ratio is that of girls to boys but then the question is about the total number of students in the classroom. That is typical of these ratio word problems, and it tends to confuse some students, especially at the beginning of their preparation period. If you are preparing for a standardized test, when you see this type of problem make sure you keep track of all the quantities involved, all the different parts as well as the total, not only the parts that come with the numbers in the given ratio. Look for the numbers the problem is not giving you. More often than not, the key to the solution is in those numbers that pertain to the situation but are not shown in the phrasing of the problem.