Problem of the Day


If a, b, and c represent real numbers, is a^7b^3c^4>0?

(1) a>0
(2) b>0

Reveal Answer

Answer

E. Statements (1) and (2) together are not sufficient.

See the Solution

Solution

[latexpage]

Statement (1): INSUFFICIENT. This tells us that $a$ is positive which means that $a^7$ is positive. ¬†However, we don’t have any information about $b$ or $c$. ¬†If $b$ and $c$ are both positive, then$a^7b^3c^4$ is positive. ¬†If $b$ is negative and $c$ is positive,¬†$a^7b^3c^4$ is negative.

Statement (2): INSUFFICIENT. This is insufficient for the same reason statement 1 is insufficient.  If $a$ and $c$ are both positive, then $a^7b^3c^4$ is positive.  If $a$ is negative and $c$ is positive, $a^7b^3c^4$ is negative.

Statements (1) and (2): INSUFFICIENT. ¬†Both statements taken together is almost enough to be sure that¬†$a^7b^3c^4$ is positive, but we can’t rule out the possibility that $c=0$, which would make¬†$a^7b^3c^4=0$ and hence not positive.


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