Vincen wrote:If \(xy>0\), which of the following must be positive?
A. \((\sqrt{xy})^2x^3y^2\)
B. \((\sqrt[3]{x})(\sqrt[5]{y})x^3y^2\)
C. \((\sqrt{xy})x^7y^9\)
D. \((\sqrt[3]{x^2y})x^2y^4\)
E. \((\sqrt{x^3y^5})x^2y^5\)
[spoiler]OA=C[/spoiler]
Source: Veritas Prep
Given that \(xy>0\), there are two possibilities: either both x and y are positive or both are negative.
Taking values of x and y as (1, 1) and (-1, -1).
Let's see each option one by one at x = y = -1. There is no need to check at x = y = 1 since when both x and y are positive, the options would also be positive.
A. \((\sqrt{xy})^2x^3y^2\)
"¢ At x = y = -1, we have \((\sqrt{-1*-1})^2*(-1)^3*(-1)^2\) = 1*(-1)*1 = -1. This is not the correct answer.
B. \((\sqrt[3]{x})(\sqrt[5]{y})x^3y^2\)
"¢ At x = y = -1, we have \((\sqrt[3]{x})(\sqrt[5]{y})x^3y^2\) = \((\sqrt[3]{-1})(\sqrt[5]{-1})(-1)^3*(-1)^2\) = -1*-1*(-1)*1 = -1. This is not the correct answer.
C. \((\sqrt{xy})x^7y^9\)
"¢ At x = y = -1, we have \((\sqrt{xy})x^7y^9\) = \((\sqrt{-1*-1})(-1)^7(-1)^9\) = 1*-1*-1 = 1. This is the correct answer.
Though we got the answer, let's try other options too.
D. \((\sqrt[3]{x^2y})x^2y^4\)
"¢ At x = y = -1, we have \((\sqrt[3]{x^2y})x^2y^4\) = \((\sqrt[3]{(-1)^2(-1)})(-1)^2(-1)^4\) = -1*1*1 = - 1. This is not the correct answer.
E. \((\sqrt{x^3y^5})x^2y^5\)
"¢ At x = y = -1, we have \((\sqrt{x^3y^5})x^2y^5\) = \((\sqrt{(-1)^3(-1)^5})(-1)^2(-1)^5\) = 1*1*-1 = -1. This is not the correct answer.
The correct answer:
C
Hope this helps!
-Jay
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