Carcass wrote:
If
a, b, c, d, and z are non-zero integers such that
(az) ^{79} * (bz) ^{71} * (cz) ^8 * (dz) ^2 > 0, which of the following must be true?
(A)
z > 0(B)
ab > 0 (C)
abcd > 0 (D)
b^2 – a^2 > 0 (E)
az + bz> 0 Kudos for the right answer and explanation
Question part of the project GRE Quantitative Reasoning Daily Challenge - (2021) EDITIONGRE - Math Book Key property: Any non-zero number raised to an EVEN power will evaluate to be a POSITIVE number. Given:
(az) ^{79} \times (bz) ^{71} \times (cz) ^8 \times (dz) ^2 > 0Rewrite as:
a ^{79} \times z^{79} \times b^{71} \times z^{71} \times c^8 \times z^8 \times d^2 \times z^2 > 0Combine all of the
z terms:
a ^{79} \times b^{71} \times c^8 \times d^2 \times z^{160} > 0Since we can be certain that
c^8 is positive, we can safely divide both sides of the inequality by
c^8 to get:
a ^{79} \times b^{71} \times d^2 \times z^{160} > 0For the same reason, we can safely divide both sides of the inequality by
d^2 to get:
a ^{79} \times b^{71} \times z^{160} > 0And, for the same reason, we can safely divide both sides of the inequality by
z^{160} to get:
a ^{79} \times b^{71} > 0Since we can be certain that
a^{78} is positive, we can safely divide both sides of the inequality by
a^{78} to get:
a ^{1} \times b^{71} > 0Since we can be certain that
b^{70} is positive, we can safely divide both sides of the inequality by
b^{70} to get:
a ^{1} \times b^{1} > 0In other words,
ab > 0Answer: B