SAT Math question help!

Test Preparation SAT Preparation
1

I was going through an SAT math review in an SAT book and I found one particularly interesting question which I got stuck on.

4^x + 4^x + 4^x + 4^x =

(A) 4^(x+1)
(B) 4^(x+2)
© 4^(x+4)
(D) 4^4x
(E) 4^x^4

I don’t understand how the answer to this is 4^x+1…

2

Here’s the trick: there are FOUR (4^x)'s.

4( 4^x) = (4^1)(4^x) = 4(1+x)
Choice A.

3

This is pretty basic algebra.

What you’ve essentially got is
1(4^x) + 1(4^x) + 1(4^x) + 1(4^x) this is the same quantity added four times
=4*(4^x) now 4 is multiplied by 4 one more time
=4^(x+1)

4

Thanks guys!

I realize this is basic algebra, but I needed to jog back my memory…

5

Also worth trying to see what happens if you just make up a number. Say x = 3…

You get 64 + 64 + 64 + 64 = 256.

Now put x = 3 into each answer choice…rule out anything that is not 256.

6

@Desi4life‌

Here is another one relevant to the SAT:

This is a grid-in one:

Given 2^n + 2^n + 2^n + 2^n = 2^9, what is the value of n ?

7

And I will be the anti-algebraist again! About 30 seconds of trial and error with a calculator will do the trick! It is not that easy to write an SAT problem that requires the kinds of algebraic manipulation you will need to do this problem the “proper” way. Not there is anything wrong with doing things the “proper” way. But good to know alternative ways as well.

8

Given 2^n + 2^n + 2^n + 2^n = 2^9, what is the value of n ?

You have FOUR (2^n)'s.
But since the base is 2, let’s call it (2^2) instead of 4.
That’s (2^2)(2^n)
The base is the same, so add the exponents
2^(2+n)= 2^9.
2+n = 9
n = 7

9

@pckeller‌

The easy and medium level problems on the SAT are most susceptible to trying numbers or substituting the answer choices. As we reach the Level 5(Hard) problems, it gets harder to plug in, not that it can’t be done. My experience is that on the harder problems, the average students struggle with the same techniques that seem to work on the easier ones. In general, I would say that if you are targeting a 750+ on SAT math, then you better know how to do things the direct way, it will save you time.

Here is an example of a problem that is from the SAT and requires algebraic skills in the context of exponent problems.

If n is a positive integer and 2^n + 2^(n+1) = k, what is 2^(n+2) in terms of k?

(A) (k-1)/2

(B) 4k/3

© 2k

(D) 2k + 1

(E) k^2

10

@SATQuantum‌

I agree that there are the occasional hard problems where algebra is the better road. But there are fewer of them than you might think, and the one you just posted is not one of them!

To do this particular problem algebraically requires a pretty sophisticated comfort level with algebra. That’s a great thing to have but not an easy thing to develop and not within the time period that most students are willing to allow for SAT prep. And yet most of my students do in fact get that one right – by making up numbers. And I find that it is quicker by far to learn this “back door play” than it is to develop the algebra chops.

For the record: make up an n value, find the k value that goes with it, find the value of 2^(n+2) using your made up n value. Then plug your k value into the answers.

Also for the record: I am not really against algebra! I even recently wrote a book to introduce algebra to middle school students. As a physics teacher, of course I think algebra is useful and important. I will be happy if the new SAT actually finds better ways to require it. But for now, for most students preparing for the SAT in its current form, the algebra can be so easily circumvented that that’s what I recommend for all but the most fluent students. I think they are the ones you are talking about as well: the ones who are realistically targeting 750+.

11

^I just noticed that plugging in even n = 1 kills all choices except for B…

I first approached the problem by simplifying it a little:

Let s = 2^n
–> 2^(n+1) = 2s and 2^(n+2) = 4s (granted, you have to apply the basic exponent properties here)

Then k = 3s, and we are asked to find 2^(n+2) = 4s in terms of k.
4s = (4/3)*3s = 4k/3

12

Quantum, no offense, but that “hard” question that supposedly requires real algebra can be done by Plugging In, as PCkeller and MIT pointed out.

Avoid algebra as much as possible. The major test prep companies say that all the time. They have millions and millions of dollars to back up their claims. If you do it the “right” way, the test is designed to waste time and trick you.

13

@mmk2015 are you sure about that? I actually believe the opposite - algebraic solutions are often (but not always) faster and more direct if you know what you’re doing. I’m sure I used a lot of algebra when taking the SAT. However, if your algebra skills aren’t great, then it might be a good idea to try these other techniques.

14

I’m inclined to agree with MITer94. Much of the time, a quick algebraic fix is faster for me than waiting for my SAT prep kids to guess and check.

The difference, of course, is that my algebra is second nature to me.

15

I mean this in the nicest way - @MITer94 I do not think you are a “typical” student. Typical students that get good grades in math probably benefit from strategies like plugging in used in place of algebra. Students that really excel, the type that enter and win competitions, should use more “real” math such as algebra.

16

You can spend 3-4 months learning how to do algebra and prob still struggle.

Or you can spend 3-4 weeks learning how to Plug In and prob improve your score quicker.

Needless to say, if you’re already scoring 700+ on the SAT, real algebra is the best route. But Plugging In works. If you know how to do it, you can tackle even the hardest questions. If you’re algebra is weak, you won’t be able to tackle the hardest questions.

17

@mmk2015 Both strategies have their (dis)advantages, and in no way am I saying that one should avoid algebra or avoid plugging in as much as possible. In general, for the SAT, any solution that guarantees you the right answer is fine.

However, I can think of at least a few hard SAT-level questions in which just plugging in stuff might not be a very good idea.

Another issue with plugging in is that it is merely a technique for the SAT (and other tests); if you wish to pursue math or any of the sciences, I wouldn’t recommend plugging in as a replacement of a solid grasp of algebra. However, I do agree that the strategy is useful at times, and can be useful even in higher-level mathematics if you want to observe the behavior of some quantity or function at certain values, etc.

18

As far as standardized tests are concerned I am neutral as to whether someone can do a problem by plugging in or by direct algebra. This includes not only the SAT but also the prestigious exams such as the American Mathematics Competition. However, we need to be absolutely clear that plugging in does not always work, and in the difficult problems (even on the SAT), it will not be easy for most students to implement. And in many cases, direct algebra will be more efficient. The reason plugging in is touted as a silver bullet by test prep companies is because it seemingly circumvents learning any “algebra”. This is what the infamous test prep companies do, look I have a magic pill that you can use and you don’t have to know any algebra and all you have to do is to plug in some simple numbers. This in essence is the appeal of the plugging in technique. In many ways this is the equivalent of a vibrating lap band that can help you lose weight while sitting on a chair all day.

I don’t think students need to be coddled and told that algebra is difficult. In my own experience, whenever I have encouraged students to hone their algebra skills, they have always enjoyed the process and have found the tests to be easier to handle. And for students who will go on to science, engineering, and finance, learning algebra is absolutely essential. The standard of algebra and math teaching in the US has declined drastically from the 1940s to the present day, and this whole business of promotion of “plugging in” technique is a mere symptom of that. If you need a drastic example of this change, just look at the [New York Regents Exam](http://www.screencast.com/t/cRGygXZVEl0) from 1944 and compare it to today’s Regents exam, which is a joke.

19

I agree with some but not all of this.

  1. If a student hopes to break 750, they should be able to apply algebra. They will only have to do that a couple of times in an entire test, but for those times, algebra will be quicker. But as students who are already scoring so high, they CAN do algebra.
  2. Students who are hoping to get into the 600s or even into the high 600s often have major algebra weaknesses. For many of them, algebra IS difficult. They are not fluent in that language, so things that seem quicker to those of us who do speak the language are in fact quite challenging for them. If the goal is to prepare them for a career in science or economics, I'd say: time to put in the work necessary to learn the language. But if the goal is to improve on a standardized test that is in a couple of months, it is silly not to learn these other approaches as well. Because...
  3. The approaches do work! That's how it is NOT like a vibrating lap band! Even as the problems begin to get hard, the "back door" of plugging in numbers is often just easier and quicker, freeing up essential time for the hardest problems at the end.
  4. As for "students who will go on to science, engineering, and finance, learning algebra is absolutely essential." -- I could not agree more! That was my main reason for writing a book to introduce algebra earlier! People who argue that algebra is not important in real life are missing this point entirely. Lack of fluency in algebra closes many doors. It is only in the narrow context of the SAT that I encourage (most) students to avoid it.
20

One would hope that, by the time someone was prepping for the SAT. it would be a matter of remembering Algebra, not “learning” it.