The Power(s) of 2

Joe Lucero —  April 25, 2013 — 4 Comments

gmat bracketEven though the NCAA tournament finished up earlier this month, for the next ten months I will be thinking about college basketball whenever I see the first several powers of two. No matter what type of GMAT question you are dealing with, our minds are better able to work through topics that we are already familiar with. Probability problems make me think of gambling, weakening a GMAT argument becomes shooting down an argument from that crazy relative you only see at Thanksgiving, and anything dealing with the number 64 comes down to rounds in a basketball tournament. Here’s a few tricks on the GMAT where knowing your powers of two can save you some time and brainpower.

 

1.  64 = 2^6

Know how to translate larger numbers into their smaller factors

Since 1985, every team that has won the NCAA tournament has had to win six games. By multiplying two times itself, you can expand to each round of the NCAA tournament- 2, 4, 8, 16, 32, and 64. And because these numbers are all small and have a single prime factor, they commonly end up on the GMAT. Because of this, you should be able to recognize them and quickly put each one into its base of two: 2 = 2^1, 4 = 2^2, etc. Same for the powers of three- 3, 9, 27, 81. The number 81 is far more likely to show up on your GMAT than 83, because 81 is a power of 3 that can be broken down into small prime factors. Without a calculator, numbers that are easy to break down show up 2 x 5 times more often than they do in the real world.

 

2.  (2^2)^3 = 64

When you’re not sure how to compute a difficult problem, think about what you would do with easier numbers

I’ve been teaching the GMAT for three years and to this day, I still have trouble translating some word problems. x is y percent more than z isn’t something that most of us can instinctively translate. But I do know that 15 is 50 percent more than 10. And when I think about why, it’s because I’m taking z and adding another y/100 of that z. Which means x = z * (y/100) + z. These equations with simple numbers make more sense to us, because we’re more likely to have to mark something up by 50 percent than to mark something up by y percent. Making analogies to things that we are used to working with makes questions easier and faster, and makes ourselves more confident in our work. Which is why when I see ((3 x 2^2)^4)^-2, I’m instantly thinking that (2^2)^3 = 64 = 2^6. If it works on a simple computation, it should work on a harder one too.

 

3.  64-1 = 63 teams that lose a game

If you get stuck answering a problem, try rephrasing the question

If I asked you how many games there are in a 64 team tournament, you might work your way through round by round: 32 games played by 64 teams in the first round, 16 games played by 32 teams in the second round, 8 games with 16 teams, 4 games with 8 teams, 2 games with 4 teams, and 1 game with the final two participants. 32 + 16 + 8 + 4 + 2 + 1 = 63. But on the GMAT, rephrasing a question can be the difference between spending three minutes on a problem and spending one. Instead of asking how many games are played, ask yourself how many games teams will lose. Think of this like the 1-x trick in probability: If 64 teams are playing and 1 doesn’t lose, that means 63 do. If you’ve taken a free-trial class with us, you know that rephrasing questions (and statements) is a huge point of emphasis in our classes beginning on day 1. So when you get stuck solving a problem, look for other parts of the question that you can solve.

 

4.  0.000064 = something to do with the number 2

When all else fails, focus on what looks familiar

Much of what makes the GMAT so difficult is the fact that there’s so much going on in any given problem and no clue about where to start. Sometimes when you look at a problem type, you will have a go to method- set up a chart or a system of equations and solve. But other times you will see a question and have no idea how to categorize it, simplify it, or do anything with it. But just because you don’t know how to solve a problem doesn’t mean you aren’t able to do a small part of the problem. Back to the very first point I made in this article- 64 is much more likely to show up on the GMAT than a prime number like 67. So when you see 64, think about how you can simplify that part of the problem. Looking at 0.000064, change the way you write that number to 64 x 0.000001 and deal with the two parts separately. Once you’ve made that step, recognize that 64 is 2^6. And once you see that one part of the problem has an exponent of 6, see if that same step can be expressed on another part of the equation: 64 x 0.000001 = 2^6 x 0.000001 = 2^6 x (0.1)^6 = (0.2)^6. GMAT problems aren’t solved with one impossible leap; they’re solved using several workable steps. Sometimes that first step will help reveal where the next step should be.

Joe Lucero

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Joe Lucero has both a Biology degree and a Master of Education from the University of Notre Dame. He also has a 780 on his GMAT. In the fall, you will find Joe in a much better mood during weeks after the Fighting Irish win their football game. During the rest of the year, you will find him looking for new places to travel, reading almost anything non-fiction, crossfitting, and trying to solve every challenge problem in the Manhattan GMAT Student Center.

4 responses to The Power(s) of 2

  1. Thanks for the good Article! Definitely a wider lens perspective and recognizing the “familiar” face of 2 will help in seemingly tough problems. Hope to remember and follow in sticky situations!

  2. In the example above in the last line of the second paragraph (#2), I see the solution to ((3 x 2^2)^4)^-2 as impossibly small. Wouldn’t this translate to (12^4)^-2 which is equal to 20736^-2 which again is equal to 1/20736^2, which equals 1/429981696. This is an impossibly small number. Am I doing something wrong?
    Can we expect such problems in GMAT?

    • Good question. The GMAT would never ask you to calculate the value of such an absurdly small fraction. However, the GMAT can (and does) ask you to know how to manipulate fractions and integers with exponents, either to find equivalent forms of a hard-to-calculate number or to cancel these numbers out, leaving some simple number.

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