Mbawarrior01
If \(x^2y=z^3\), is \(z^3 > 0\)?
(1) \(xy^2 > 0\)
(2) yz > 0
We are given that (x^2)y = z^3, and we need to determine whether z^3 > 0. Recall that the values of z^3 and z have the same sign; thus, we need to determine whether z > 0.
Statement One Alone:x(y^2) > 0
Since y^2 is always nonnegative, x(y^2) > 0 means x > 0. However, since we cannot determine the sign of y, we cannot determine whether z^3 = (x^2)y is greater than 0.
For example, if y > 0, then z^3 > 0; however, if y < 0, then z^3 < 0. Statement one alone is not sufficient to answer the question. We can eliminate answer choices A and D.
Statement Two Alone:yz > 0
Since yz > 0, we know that either both y and z are positive or both y and z are negative. However, since we cannot determine the sign of y, we cannot determine whether z^3 = (x^2)y is greater than 0.
For example, if y > 0, then z > 0, and hence z^3 > 0; however, if y < 0, then z < 0, and hence z^3 < 0. Statement two alone is not sufficient to answer the question. We can eliminate answer choice B.
Statements One and Two Together:Using the two statements together, we still cannot determine the sign of y and thus cannot determine whether z^3 > 0.
Answer: E