# An Exponent Problem

Consider the following question:

$x=2^{y}-(8^{7}-8^{5})$

Which of the following values of y is closest to 0?

A) 24

B) 21

C) 20

D) 16

E) 14

Here’s a question designed for a calculator, you might say. But let’s set the calculator down for a moment and break this down. Realize that a problem like this is designed to be solvable. Let’s look at it step-by-step.

Step 1: Ask yourself how could this be possible? It’s a useful question to ask whenever you encounter a difficult problem, whether inside the test prep world or outside, but it’s an especially useful question to ask in a situation such as this where you know there is some possible and not extremely difficult solution. Here, the answer to the question is that this must be true:

$2^{y}-2^{y}=0$

That’s our path toward a solution.

Step 2: Make the information you have look as much like your projected solution as possible. How do we make a base of 8 into a base of 2?

$8^{7}=(2^{3})^{7}=2^{21}$

So, we can start to break down the second piece of the equation into:

$8^{7}-8^{5}=2^{21}-2^{15}$

Step 3: Analyze. Now we must find what value of $2^{y}$ is closest to the value in the equation above. Again, you might be tempted to pull out a calculator, but resist the urge. $2^{21}-2^{15}$ appears to be an ugly jumble of numbers and symbols. What does it mean? Think about what powers of 2 mean. It means $2^{21}$ is twice as big as $2^{20}$ which is in turn twice as big as $2^{19}$ and so on. $2^{15}$ is only 1/64th as big as $2^{21}$ so subtracting it out doesn’t move us very close to $2^{20}$$2^{21}$ is still the value of y that gets us closest to 0, and B is your correct answer.

# A Note on Perspective

I know that we spend a lot of time here talking about improving your scores as much as you possibly can and pushing small advantages as much as you possibly can. But sometimes it pays to get a little perspective.

I was recently over at my parents’ house cleaning out the last of my old stuff from the garage. I came across the printout of my PSAT results. This immediately triggered two thoughts:

1. I can’t believe I saved this

2. I wonder what I got

That second thought is the one I want to focus on right now. It’s interesting that someone who is immersed in the test-prep industry doesn’t even know what he got on the PSAT. And that’s completely okay. That score and that number didn’t define my life. It was merely a means to an end. A stepping stone toward the next phase of life. Fifteen years later those results have become irrelevant in my life.

I have some students who obsess about every question, every point, every wrong answer. Sometimes you just need to step back and take a wider view. Absolutely you should make every effort to do your very best work. But occasionally you need to consider what’s really going to matter 10 years down the line.

Keep working hard, but keep perspective!

# Naming the Numbers

When getting back to math-based questions after it’s been a while since math class, many of us need a refresher on some key terms that are used and what they mean. Here are five that you’re likely to come across and a quick definition to refresh your memory.

1. Integer- Probably the most commonly used number term, integer is a commonly confused term. I like to say that if you were going to draw a number line, an integer is anywhere you would draw a dash, including zero and negatives. So, -2, -1, 0, 1, 2, and 3 are all integers, while 1.4 and 2/3 are not.

2. Whole Number- While this term isn’t commonly used in tests, it’s one you’ve heard before and it can easily be confused with the term integer. While there is a great deal of overlap between the two groups, the group of whole numbers does not include negative numbers. So, while 0, 1 and 2 are both integers, -3 is an integer but not a whole number.

3. Multiple- A number is a multiple of another number– for instance 8– if it is the product of that number and an integer. So, 24 is a multiple of 8 because 24= 8 * 3 and 3 is an integer. Similarly, 0 and -16 are multiples of 8 because they are the product of 8 and an integer (8 * 0 = 0, 8 * -2 = -16). 4 is not a multiple of 8 because it is the product of 8 and a non-integer (8 * 0.5 = 4).

4. Factor- A factors are the integers you multiply together to get a number. They come in pairs. For instance, if we were  factoring 18 we would see that 6 * 3 = 18, so both 6 and 3 are factors. So are 9 and 2, and 18 and 1. Factors can be negative as well, although that is rarely tested.

5. Prime Numbers- A prime number is a number whose positive factors are only 1 and itself. Since no integers other than 1 and 13 divide evenly into 13, it is prime. Common misconceptions about prime numbers are than 1 is prime (it isn’t), and that all prime numbers are odd (2 is a prime number).

Hopefully that served as a good refresher!

# You’re Smart, But…

I was watching American Ninja Warrior with my wife the other night. It’s the American version of a Japanese show that has contestants race through a ninja-inspired obstacle course. The obstacles require incredible amounts of upper body strength, balance, agility, grip strength and creativity. No one has been able to complete all four stages of the course over the several seasons the American version has been on the air.

The difficulty of the task and the possibility of big prize money attracts competitors from all walks of life. From the enthusiastic fan, to the experienced rock climber to the gym owner or the professional free runner, they all bring their experiences and hopes to the course.

In a recent episode, some of the top rock climbers in the world came to take a shot at the course. These are guys that have all of the physical tools that it takes to complete the course. They have amazing upper body and grip strength, and the kind of agility and athleticism it takes to be world-class. And yet, they struggled on some areas of the course.

Why?

Even though these athletes had amazing skills, they hadn’t yet adapted those skills to the particular techniques and challenges that the course put in front of them. Sure, rope climbers spend a lot of time on ropes, but the rope jungle maze was trouble because they weren’t used to ropes that moved or dropped in the certain ways that these ropes did.

Standardized tests like the ACT, SAT, GRE and GMAT offer their own types of rope jungles. You can have every single skill that would be necessary to conquer the test, but if you don’t have the particular techniques necessary that might not be enough.

You’re smart, but are you ready for this jungle?