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If x and y are integers, is x^7 < 6^y?

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Source: — Data Sufficiency |
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by Vincen » Mon Apr 22, 2019 7:00 am

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If x and y are integers, is x^7 < 6^y?

(1) x^3 = -125
(2) y^2 = 36

[spoiler]OA=A [/spoiler]

Source: Princeton Review

Hi M7MBA.

Let's check the statements.

Statement 1:
(1) x^3 = -125
Here we are told that x=-5. Now, $$x^7=\left(-5\right)^7=-78125.$$ On the other hand, notice that 6^y will always be positive and since x^7 is negative, then x^7 < 6^y. So, this statement is SUFFICIENT.

Statement 2:
(2) y^2 = 36
This statement tells us that y=-6 or y=6 but this is not enough information to give an answer. Because in any case (y=-6 or y=6) we can pick values of x for which the inequality is true and false. Hence, this statement is NOT SUFFICIENT. $$y=6\ and\ \ x=1\ \ \Rightarrow\ \ \ x^7<6^y\ is\ true.$$ $$y=6\ and\ \ x=7\ \ \Rightarrow\ \ \ x^7<6^y\ is\ false.$$ In conclusion, the correct answer is the option _A_.

I hope it helps.
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