Each of the following equations has at least one solution

[tim]

It doesn’t matter. Since this is an EXCEPT question, we’re just trying to find values of n that cause each expression to fail. Without any restrictions on n, we are of course free to test odd numbers, even numbers, or non-integers..

[ffearth]

.

[tim]

because using a calculator is definitely a "math error" on the GMAT - one that will get you disqualified!

[mailsunild]

Hi, spent a lot of time to decipher, but could not. I picked 1 & 2 and checked for each of the options and got the below:

The Question as well as each ans is confusing, for eg. - how can 1/2 = -2 in result (B) below (or is my understanding itself is wrong?)?

If n=1:
A) -2 = -1/2
B) 1/2 = -2
C) 2 = -1/2
D) -2 = -2
E) -1/2 = -1/2

If n=2:
A) -4 = 1/4
B) 1/4 = 4
C) 4 = 1/4
D) 4 = -4
E) 1/4 = -1/4

Can someone explain please.

Thanks
Sunil

[tim]

when you get things that don't equal, that means the specific number you plugged in doesn't work for that equation. take a closer look at what they're asking though: because this is an EXCEPT question, you need to find four answer choices that actually have solutions (i.e. the equation works). anytime you find one where the equation works, that WON'T be the answer. get rid of the four wrong answers and what's left is the right answer. try plugging in 0 and that will help..

[BekermanLeo]

So, I logically understand why A cannot ever have a solution, but that means I am making a conceptual algebra error. Here was how I tackled A specifically:

-2^n = (-2)^-n

-2^n = -1 / 2^n

-2^n * 2^n = -1

- 2^2n = -1

2^2n = 1

So, n = 0 would be valid in that case. When you flip an expression with a negative exponent, are you supposed to keep whatever is inside the parentheses?

[RonPurewal]

So, n = 0 would be valid in that case. When you flip an expression with a negative exponent, are you supposed to keep whatever is inside the parentheses?

Whatever is raised to the power is still raised to the power.

so, (-2)^-n would be 1 / (-2)^n.
You could split that into 1 / [(-1)^n * 2^n].
This is where you have a problem: (-1)^n is only -1 if n is odd. If n is even, it's 1.

There's nothing special about -1, by the way. If you don't understand why these steps work the way they do (or, more pointedly, why they don't work in the way you originally tried), then just try it with 2*3 = 6 instead of 2*-1 = 2:
6^-n
= 1 / (6^n)
= 1 / [(2^n)(3^n)]
Same thing, with "-1" in place of "3".

[RongP360]

For answer choice D and E, how do we know that those are absolute values? I must be missing something here.

Thanks a lot!

[RongP360]

To add to my earlier post, below is the explanation provided by the CAT solutions

"(D) The left side is positive for even values of n and negative for odd values of n, while the right side is always negative; the absolute values of the two sides are always the same (= 2n). Therefore, any odd value of n will solve this equation.

(E) The left side is positive for even values of n and negative for odd values of n, while the right side is always negative; the absolute values of the two sides are always the same (= 2-n). Therefore, any odd value of n will solve this equation. "

[RonPurewal]

don't think of that as "absolute value". just think of that as "the SIZE of the number".

try taking some different values and plug them in for "n", and you'll see what that is all about.
for D and E, the numerical value is always the same; the only difference is whether the number is positive or negative. (in the first three choices, on the other hand, the two exponents have opposite signs -- so, the SIZES of the numbers are vastly different for most values of n.)
that's the point there.

you're not supposed to be using "absolute value concepts". the point there is just that D and E are the same SIZE, but that they might have either the same sign or opposite signs.

[ZairaA313]

Kiran,
This is a tough question, which is why it is a 700-800 level question. You have to focus on a few important aspects:
1) pay attention to parentheses, as Ben demonstrated above.
2) make sure you know how to deal with a negative power--for example, know that 5^-2 = 1/25. If you have a negative power you have to take the original number's reciprocal then apply the exponent to the entire new number.

For me the easiest way to do this problem is to think "what are some numbers I can substitute in for n that are easy to work with and may allow me to cross off some answers?" For me what comes to mind is n=0 or n=1. Anything to the 0 power will be 1, whereas anything to the 1 power stays the same. By plugging in 0 for n I can eliminate B and C; by plugging in 1 I can eliminate D and E. Trying plugging those in and writing the equations out. You can probably see why these solutions do work and why those answer choices must be eliminated.

A is the one that does not work for any value of n. On the left side of the equation, we are taking 2^n and then applying a negative sign. That result will always be negative. On the right, I can only get a negative result if I use an odd integer for n. However, my result will be a reciprocal of what is on the left.

For example, if n = 1, I get -2 on the left and -1/2 on the right.
If n=3, I get -8 on the left and -1/8 on the right.
And so on.

Thus, an even n doesn't work because I'll get a negative on the left and a positive on the right. An odd n doesn't work because I get the reciprocal situation described in the paragraph above.

A is the only one without a solution for n.

Great explanation. Thank you!

[RonPurewal]

may i ask why you're responding to a post that is almost six years old?