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26 May 2017, 04:42
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If y is an odd number and \(x^{3}y^{4} = 648, y=\)?
(1) \(y\) is positive.
(2) \(x\) is an integer.
(1) \(y\) is positive.
(2) \(x\) is an integer.
26 May 2017, 04:42
Kudos
Bookmarks
Official Solution:
You can take the 1st step of the variable approach, and modify the original condition and the question.
\(648 = (2^{3})(3^{4})\) (1)
\(= (2^{3})(-3)^{4}\) (2)
\(= (8)(3^{4}) = (1^{3})(\sqrt[4]{8}\times 3)^{4}\) (3)
\(= (1^{3})(-\sqrt[4]{8}\times 3)^{4}\) (4)
\(= (2^{3})(3^{3})(3^{1}) = (6^{3})(3^{1}) = (6\times \sqrt[3]{3})^{3}(1^{4})\) (5)
\(= (6\times \sqrt[3]{3})^{3}(-1)^{4}\) (6)
You get the results as shown above. In other words, if it is a negative number, you must remember to consider the irrational numbers and to find a hidden 1. Then,
in the original condition, y is an odd number, so you get (1), (2), (5), (6), and there are 2 variables \((x, y)\) and 1 equation \((x^{3}y^{4} = 648)\). In order to match the number of variables to the number of equations, there must be 1 equation. Therefore, D is most likely to be the answer.
In the case of con 1), if \(y > 0\), from (1), (5), \(y = 3,1\), hence not unique and not sufficient.
In the case of con 2), if \(x = integer\), from (1), (2), \(y=3, -3\), hence not unique and not sufficient.
If so, by solving con 1) and con 2), you only get (1), and \(y = 3\), hence it is unique and sufficient. The answer is C.
Answer: C
You can take the 1st step of the variable approach, and modify the original condition and the question.
\(648 = (2^{3})(3^{4})\) (1)
\(= (2^{3})(-3)^{4}\) (2)
\(= (8)(3^{4}) = (1^{3})(\sqrt[4]{8}\times 3)^{4}\) (3)
\(= (1^{3})(-\sqrt[4]{8}\times 3)^{4}\) (4)
\(= (2^{3})(3^{3})(3^{1}) = (6^{3})(3^{1}) = (6\times \sqrt[3]{3})^{3}(1^{4})\) (5)
\(= (6\times \sqrt[3]{3})^{3}(-1)^{4}\) (6)
You get the results as shown above. In other words, if it is a negative number, you must remember to consider the irrational numbers and to find a hidden 1. Then,
in the original condition, y is an odd number, so you get (1), (2), (5), (6), and there are 2 variables \((x, y)\) and 1 equation \((x^{3}y^{4} = 648)\). In order to match the number of variables to the number of equations, there must be 1 equation. Therefore, D is most likely to be the answer.
In the case of con 1), if \(y > 0\), from (1), (5), \(y = 3,1\), hence not unique and not sufficient.
In the case of con 2), if \(x = integer\), from (1), (2), \(y=3, -3\), hence not unique and not sufficient.
If so, by solving con 1) and con 2), you only get (1), and \(y = 3\), hence it is unique and sufficient. The answer is C.
Answer: C
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