Data Sufficiency

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 ?

(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.

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 ?

(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.

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