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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