# 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.

# 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!

# Bloody Lips and Second Chances

I had my biggest parental fail yet last week. I was in the process of unbuckling my son out of his high chair when I got distracted and turned my head. When I turned back, he’d tumbled forward and taken a spill. He and I were both pretty shaken up by the whole experience, but he’s on the mend and will be just fine.

In the days following, I’ve been a little paranoid and extra careful. But I wonder: had I left him in that dangerous position before? It could be that this was the first lapse I’d had and it turned out poorly, or it could be that I’d made this mistake before and gotten lucky because nothing bad had happened.

I don’t have a nanny cam running in my home 24-7 to monitor my parental performance, so I can’t really know the answer to that question. But if I did, this accident might not have happened. There’s a benefit to being able to review your performance, even if the end result turned out just fine.

Let’s carry that lesson over to the test prep world. Sure, it makes sense to spend most of your review time going over problems you missed so that you’re better able to understand what went wrong and how you can do better next time. However, it does make sense to spend some time going over the questions you get correct as well. At minimum it can reinforce good habits, and potentially you may spot mistakes that didn’t end up costing you a correct answer on this problem. However, you may not be so lucky next time.

For instance, if asked for the square root of 9, and given an answer choice of 3 you won’t be penalized for forgetting that -3 was also an option. However, if you review that correct answer and realize your mistake you’re less likely to make the same mistake where your error is the difference between a right answer and a wrong one.

As I’ve seen this week it’s helpful to learn from a wrong choice, but it’s even better if you can learn without suffering the consequences.

# A Word of Caution When Plugging in Numbers

The positive numbers m and p are variables in the following equation:

4m + 5mp + 10 = 50m

What is p in terms of m?

A) 46/5 m

B) 9m + 5

C) 46/5 – 2/m

D) 5/m + 2

E) m/5 + 46

There’s always going to be an algebraic solution to this problem, but this one is… a little ugly. The solution doesn’t come out neatly, so there’s a fair amount of potential to make a mistake or lose confidence as you’re working through it. I hope you respond to that by saying, “Don’t worry! I see variables in the answer choices, so I can just plug in my own numbers!” That’s a wonderful thought, but here is how that might go wrong if you try it.

“Well, I know I’m supposed to pick numbers that are easy to calculate, so let me just use m = 2 and p = 3. If we plug those in to our original equation we get:

4(2) + 5(2)(3) + 10 = 50(2)

8 + 30 +10 = 100

48 = 100

Huh???

When plugging in your own numbers into a problem, it’s important to know that your numbers need to follow the rules set up in the problem. But don’t worry, that doesn’t mean you need to magically pick a pair of numbers that works out of the air. Notice, in this case we don’t see both variables in the answer choices, only m. So, let’s pick a value for m, plug that into the equation and see what value it spits out for p.

4(2) + 5(2)p + 10 = 50(2)

8 + 10p + 10 = 100

10p = 82

p = 41/5

So now we have a pair of numbers that works for this equation. We know that when m=2, p=41/5. Now we can just test that pair against our answer choices to see which equation gives us the value of p that we found.

A) 46/5 m; 92/5… Nope

B) 9m + 5; 23… Nope

C) 46/5 – 2/m; 41/5… YES

D) 5/m + 2; 9/2… Nope

E) m/5 + 46; 232/5… Nope

Plugging in numbers can be a valuable strategy, but make sure you’re using it correctly!