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.

Negative exponents

Using powers of two for explaining the concept of negative exponents

Negative numbers confuse many students. Usually students tend to struggle with almost everything related to negative numbers. From the very concept of using the (-) sign to refer to conventional spatial directions (left, down, back), to the different rules for adding, subtracting, or multiplying positive and negative numbers; there are plenty of instances where the (-) sign is overlooked, or misinterpreted, resulting in a wrong answer. Given that a high number of wrong answers produce a low score, it is only natural for many students to react defensively whenever a new concept involving negative numbers shows up in their radar. This is the case for negative exponents. Remember that the first definition of exponent (positive) most students are introduced to is: “the number of times you multiply a number times itself.” When one tries to apply this definition to the case of negative exponents it does not make sense because, how do you multiply a number times itself “minus once,” or “minus twice”? One possible answer is that, when it comes to negative exponents, you do not multiply but you divide instead. Even with that interpretation the rule still needs some adjustment because, if you start with a number a, and you divide it by itself once, you get 1 as a result, so you would think a^-1 was 1 but that would be wrong because 1 = a⁰, whereas a^-1 = 1/a.
In order to avoid this kind of confusion, I follow another approach: By asking very simple questions I help the student build a list of powers of 2 from 2 to 1024, each related to their corresponding positive exponent. Then we notice the patterns how the numbers change, and how the exponents change. Then we move backwards all the way to the beginning of the list, and beyond, continuing into fractions, on one column, and into negative exponents, on the other column. In this way it becomes crystal clear to the student how the exact same pattern correlates the negative exponents to the fractions, and even the general formula a^-n = 1/aⁿ becomes apparent. I have successfully used this method many, many times, and so far it has worked wonderfully, clearing away all confusion and anxiety students of all ages had around negative exponents. The key to this approach is being very thorough, and asking the right questions, in the right way, at the right time.

Lowest Common Multiple

A Method and an Analogy to Clarify this Topic

During the last few weeks I came up with a way to explain how to calculate the lowest common multiple (LCM) of two or more integers or two or more polynomials.
Many students get confused by this LCM topic. One reason is the simplicity of the fact that, for any two expressions, their product is a common multiple, so, “Why look any further?” many students ask themselves.
They know their teacher told them in general the product is not the lowest common multiple of two expressions, so they know they are going to get marked down if they give that answer, but many do not know how to find the LCM.
Recently I improved my success rate at explaining how to find the LCM when I started using a table format, as follows.
In the head row I write the two or three polynomials or integers for which we are looking their LCM.
On the left margin I make a list (going down) of all prime factors of the expressions involved, without any repetition. Common factors get listed just once regardless of how many expressions they appear in.
Then, having one row per factor, and one column per expression, we fill in the table by writing the exponent each factor appears raised to in each expression, carefully including all exponents (even those with value zero or one).
After all exponents are listed in the table we make another column at the far right, under the heading “Maximum.” There we write the biggest number out of each row.
The next step is to form the LCM as the product of all individual factors listed in the table (in the leftmost column), each raised to its maximum exponent, as listed in the rightmost column (under “Maximum”). This last product is the LCM we were looking for.
Of course this process can be done without the table, but the table makes it explicit, and it helps as a visual aid for the student to see everything that is going on, all at once. It also helps in making very clear that we do not add the exponents, nor do any other operation with them, we only identify and select the biggest one for each factor.
Most students are happy with this process; the table is good enough for them. It gives them a clear method to follow, and it takes away the guessing and the mystery they formerly faced when trying to calculate the LCM. One of them even said: “You just saved my life with that table! Now I know how to do it!”
However, there are always a few students who also want to know why the procedure works, not only how to do it.
For those who ask “Why?” after seeing the table, I have this explanation ready:
“We have to imagine we are watching a movie about spies and intelligence agents, O.K.? Each expression is like a security checkpoint, where our agent has to show the proper clearances to get pass that point. The checkpoints have different sets of requirements. Each requires verification of a certain level of authority for each security category they are checking at that point. The factors of the expressions are the security categories, like “radioactive material,” “fire arms,” “chemical hazards,” and so on. The exponents are the different levels of clearance agents may have in each category. So when determining the LCM we are looking for the bare minimum possible set of clearance levels we need to give an agent for him or her to be able to make it through all the checkpoints, without any extra, unnecessary authority. They don’t lose their credentials when they go through a checkpoint. They only need to show their badges, they do not give them up. That is why we do not need to add exponents; we only need to select the highest from all the expressions for that particular factor.”
I have found this explanation works very well with all students with whom I have used it so far. One of them said: “Oh! I see. The x² from 3x²y is already included in the x³ from 5x³(x+1) because the exponent 3 is higher than 2. We do not need x⁵ or x⁶. Just x³ will be enough.” And I said: “That is exactly how it works!”