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BTG practice question - DS

by garry123 » Sat Sep 11, 2010 5:26 pm
Please let me know if this is not allowed. I won't post questions from BTG practice test if it is not allowed. But I felt something wrong with the question.

If x ≠ 0, what is the value of Image ?

(1) w = y

(2) x^2 = 1

OA D

But my answer and explanation are [spoiler]A - Reason for statement 2: It not no where mentioned that w and y are integers, if w=5/4 and y = 2/4 => w-y = 3/4 and the final answer will be (-1)^3, but when w and y are integers the answer will be always 1.[/spoiler]
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by OliverTPR » Sat Sep 11, 2010 6:10 pm
I am not sure I follow your logic in your explanation. I think you may be missing a number or a word. Here is what I take from the problem.

x^2 tells us that x is either -1 or 1. Given that this is true, x^w is either 1 or -1 and x^y is either 1 or -1. So, no matter what happens x^w/x^y is either 1 or -1. Now, since this value is then taken to a positive root, the negative, if present, will become a positive one. So no matter what w and y are, the answer will be 1.

Now, if you are concerned that w could be 1/2 for example, and that we could be then faced by the prospect of a (-1) ^1/2, which is imaginary, then you are correct. The problem should have specified that w and y were integers. Is that what you meant?

Of course, imaginary numbers raised to the 4th power equal 1 as well, so I guess in either case it all comes out as 1. Still, the actual GMAT does not write questions in this manner.
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by garry123 » Sat Sep 11, 2010 7:37 pm
If it is not given W and Y are integers then 2nd statement is not sufficient.

you can write (-1) ^2 as (-1)^[(x)*4] where x is 1/2. Now you can not say we can not take -ve square root because when we will expand the powers it will be nullified and the final numbers will be real.


The only catch in this question is non existence of any information regarding W and Y.
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by garry123 » Tue Sep 14, 2010 2:16 pm
bump !!
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by Ian Stewart » Tue Sep 14, 2010 4:19 pm
garry123 wrote:Please let me know if this is not allowed. I won't post questions from BTG practice test if it is not allowed. But I felt something wrong with the question.

If x ≠ 0, what is the value of Image ?

(1) w = y

(2) x^2 = 1

OA D

But my answer and explanation are [spoiler]A - Reason for statement 2: It not no where mentioned that w and y are integers, if w=5/4 and y = 2/4 => w-y = 3/4 and the final answer will be (-1)^3, but when w and y are integers the answer will be always 1.[/spoiler]
You are perfectly correct - the question needs to specify that w and y are integers; otherwise Statement 2 is not sufficient. In fact, Statement 1 is not sufficient either, since if the exponents might be, say, 1/2, and x might be negative, the entire expression could be undefined.

As a technicality, for interest only (not important for the GMAT, since the GMAT will always be sure that any algebraic expressions are properly defined) : if, as in your example, we let the exponent y be equal to 1/2, and we let x = -1, then the entire expression is undefined (it is not equal to -1), since by order of operations, we'd first need to calculate the value of x^y, which would be equal to (-1)^(1/2), and that number doesn't exist, at least in the GMAT world where all numbers are real numbers. The fact that we later raise this to the power of 4 does not matter here; when you raise a number that 'doesn't exist' to the power 4, it cannot suddenly begin to exist again. In the same way, a fraction like 4/(5/0) is not equal to zero, even if you might, using your fractions rules, rewrite it as 4*(0/5); it's undefined, because 5/0 is undefined.
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by Brent@GMATPrepNow » Wed Sep 15, 2010 8:40 am
Ian makes a great point, and we'll pull the question and reword it.

This actually brings up a question that I've wondered about for a while now.

Here's the setup: We know that we can rewrite x^(a/b) in 2 ways: 1) the ath root of (x^b) or 2) [the ath root of x]^b

Now we know that (-2)^3 = -8
But if we rewrite 3 as (6/2), we get a problem using either of the above forms of x^(a/b). We get:

(-2)^3 =(-2)^(6/2) = sqrt[(-2)^6] = sqrt[64] =8
Or
(-2)^3 =(-2)^(6/2) = [sqrt(-2)]^6 = impossible since we can't find the square root of -2





Ian Stewart wrote:
garry123 wrote:Please let me know if this is not allowed. I won't post questions from BTG practice test if it is not allowed. But I felt something wrong with the question.

If x ≠ 0, what is the value of Image ?

(1) w = y

(2) x^2 = 1

OA D

But my answer and explanation are [spoiler]A - Reason for statement 2: It not no where mentioned that w and y are integers, if w=5/4 and y = 2/4 => w-y = 3/4 and the final answer will be (-1)^3, but when w and y are integers the answer will be always 1.[/spoiler]
You are perfectly correct - the question needs to specify that w and y are integers; otherwise Statement 2 is not sufficient. In fact, Statement 1 is not sufficient either, since if the exponents might be, say, 1/2, and x might be negative, the entire expression could be undefined.

As a technicality, for interest only (not important for the GMAT, since the GMAT will always be sure that any algebraic expressions are properly defined) : if, as in your example, we let the exponent y be equal to 1/2, and we let x = -1, then the entire expression is undefined (it is not equal to -1), since by order of operations, we'd first need to calculate the value of x^y, which would be equal to (-1)^(1/2), and that number doesn't exist, at least in the GMAT world where all numbers are real numbers. The fact that we later raise this to the power of 4 does not matter here; when you raise a number that 'doesn't exist' to the power 4, it cannot suddenly begin to exist again. In the same way, a fraction like 4/(5/0) is not equal to zero, even if you might, using your fractions rules, rewrite it as 4*(0/5); it's undefined, because 5/0 is undefined.
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by Ian Stewart » Wed Sep 15, 2010 10:09 am
Brent Hanneson wrote:
This actually brings up a question that I've wondered about for a while now.

Here's the setup: We know that we can rewrite x^(a/b) in 2 ways: 1) the ath root of (x^b) or 2) [the ath root of x]^b
This definitely not important for the GMAT, but that rule is only strictly true when x is positive (you need to extend to complex numbers if you have fractional exponents and negative bases) which is why your examples lead to some strange results. There's a more detailed discussion in the section 'Negative nth roots' here:

en.wikipedia.org/wiki/Exponentiation
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