Here’s a question I love to throw at students early on in the tutoring process (let’s call this a grid-in for now, to keep things simple):

1. If $\fn_phv&space;\frac{4^{999}+4^{998}}{5}=4^x$, what is x?

It’s a beautiful question because no matter what, it’s going to show me something about the kid with whom I’m working. Almost everyone goes to the calculator first. Once it becomes clear that the calculator will be no savior I see a few divergent paths, all illuminating:

1. If my student says it can’t be done, I know one kind of question on which I’m going to have to drill her repeatedly.
2. If my student says x = 1997, then I know he just added the exponents in the numerator and completely ignored the denominator, so we’re going to need to review the exponent rules and get his vision checked.
3. If my student factors 4998 out of the numerator to see that everything else cancels out and x = 998, then I know I’m going to have to really challenge her to get her score higher than it already is (full solution explained at the bottom of the post).
4. And if my student starts wrestling with other, more manageable numbers (for example: $\fn_phv&space;\frac{4^{3}+4^{2}}{5}=16=4^2$),  to try to suss out a pattern, I know I’m dealing with a kid who knows how to struggle and who doesn’t back down from tough questions.

Let’s be clear here: it’s best to know how to do this question the right way, like the third student. She has a strong base of math knowledge, has seen enough similar problems not to misapply exponent rules, and is creative enough to try pulling out the greatest common factor to see if anything good happens (and it does). But I don’t know yet what she’s going to do when she gets to a problem that’s unlike any she’s seen before (and on the SAT, that will indubitably happen, and probably when it counts). So I’m going to keep watching her closely until I get to see how she reacts to a question that makes her squirm.

The fourth student is one who finds a way to claw out the correct answer when faced with an intimidating problem that his tools seem at first not to be able to solve. He might not be as conversant with math as the third student, but in the eyes of the SAT, she and he are exactly the same on that question. Because he’s scrappy. He’s nimble. And that will take him a very long way.

In sports, you’ll often hear a commentator say, “That’s why you play the game,” after an underdog wins a game it shouldn’t have. It doesn’t matter who looks better on paper. It matters who performs on game day. After seeing him work this question, I’m going to worry less about student #4 on game day.

If you want to take your place in the pantheon of great test takers, you’re going to have to be nimble. You’re going to have to grapple with tough problems sometimes. You’re going to have to, as poor Mario does in the video above, make mistakes, learn from them, and try not to repeat them. You’re going to have to be flexible, and willing to try more than one approach.

This is why I want you to know how to plug in, but I also want you to be able to do the math. This is why I want you to have a decent essay skeleton in mind when you walk into the test center, but I also want you to be able to handle a fakakta prompt like the one about reality TV. This is why it’s important to have a good vocabulary, but you’re wasting your time if you think the path to a higher reading score is through flash cards or word lists alone.

Kids sometimes complain that I ask too much of them. But the way I see it, I really only want one (admittedly multifaceted) thing out of my students: I want them to learn to be nimble.

##### Solution to the sample problem:
$\fn_phv&space;\frac{4^{999}+4^{998}}{5}=4^x$
$\fn_phv&space;\frac{4^{998}(4+1)}{5}=4^x$
$\fn_phv&space;\frac{4^{998}(5)}{5}=4^x$
$\fn_phv&space;4^{998}=4^x$
$\fn_phv&space;998=x$

I love this question. Took me a while till I realized the ratio of 4^n to 4^n-1 is just 4. Did you make this question yourself? If so, you should help the makers of the SAT haha.
I like the post too; it’s scary though. I used to be like student 4 until I began high school. Now I’m like student 3 and assume I should be able to do anything correctly with rules I know.

sarah bliety says:

Great Advice! I think I’m student 1 because that’s the first thing that came to my mind. Calculator. But I’m trying to change that and hopefully I will. I just don’t understand the question why did you change 4^999 to (4+1) ???? Thank you!

Hi Sarah! Glad you’re here, and hope I can help you make that change. 4^999 doesn’t change to (4+1). Look closely…that step also loses a + sign. What’s happening there is I’m factoring a 4^998 out of the expression 4^999 + 4^998.

(4^999)/(4^998) = 4
(4^998)/(4^998) = 1

That’s why it’s (4^998)(4+1)

Does that help?

Amy says:

I have the same problem as Sarah. I’m having trouble visualizing the example and I cant check on my calculator either (sorry its instinctive). I thought that if you’re dividing two like bases with two unlike exponents then you just subtract the exponents. Then why does (4^999)/(4^998) equal 4?? Shouldn’t it equal 1 since
(4^999)/(4^998) = 4^(999-998) =4^1. Im confused =/

🙂 Sorry, for some reason I didn’t see your first comment come in! Glad you get it now.

Nikki says:

Hi sorry, I still really don’t get how that’s factoring :S You just divide the first number by the second number to facter? And how does that give you four? Unless your saying you subtracted the second number from the first and that gave you 4^1…? But then why did you divide the second number by itself again….? Sorry, would you mind explaining it step by step? So confused! 🙁

Think of it with smaller numbers, maybe. If I had x^3 + x^2, that’s the same as (x^2)(x + 1). That’s factoring x^2 out of the expression.

That’s all that’s going on above. 4^499 + 4^498 is the same as 4 × 4^498 + 4^498, which is the same as 4^498(4 + 1).

Am I helping at all?

Nikki says:

Lol took me a second but finally got it with the smaller numbers! Never would of thought of doing that… Guess I just need to practice techniques! Thanks for the help though!! 🙂

Jack Parker says:

Could you also take the log of both sides, change the right side of the equation to xlog4, and then divide both sides by log4 in order to solve for x? The calculator won’t complete the calculation due to an overflow error, but I believe that this solution is correct, just not in the form of an integer. [log(((4^999)+(4^998)) / 5))] / log4

Lily Zhong says:

i don’t really understand how (4^999 + 4^998)/5 = 4^x became 4^998(4+1)/5. can you explain it? or better yet, can you explain how to factor exponents?

It might help to think in terms of smaller exponents to start. Let’s look at 4^5 + 4^4. You can rewrite that in long form:

(4)(4)(4)(4)(4) + (4)(4)(4)(4)

Now you can factor out as many 4s as both terms share. Remember, when you factor an entire term out of itself, you have to leave a 1 in there.

= (4)(4)(4)(4)[(4) + 1]

= 4^4(4+1)

Does that help at all?

Nick Gomez says:

To be honest, I’m more of a student 4 even though I solved the problem like student number 3. I think logically and try small scale things when I can’t seem to solve a problem. Fortunately, I’ve used (anti-?) distribution so many times that problems involve it just discern themselves to me.

Anonymous60 says:

solving this with a calculator is very,very quick and much more visible (for most students) than the solution by student#3.. If you just simplify the first expression with a calculator and then take the “log base 4” of both sides, you will have x isolated…

deniz says:

I’m a student three mostly because I don’t use calculator even when I have to… I somehow keep forgeting I’m allowed to use a calculator probably because we don’t use it even in calculus in my country. I end up missing a question or two and they are always questions could be solved with a calculator.