Grades 11-12 Video Solutions 2022-2022_11-12

Video Transcription

Let a, b, c be non-zero numbers. The numbers negative 2 a to the fourth b cubed c squared and 3 a cubed b to the fifth c to the minus four have the same sign. Which of the following  is definitely true? So the main idea for this problem is that square numbers are positive and don't affect  the sign at all. So we can just ignore them essentially. So for this number we can rewrite  it as negative 1 times 2 times a to the fourth b squared times b times c squared and we can ignore the 2 ignore the a to the fourth and ignore the b squared and ignore the c squared  and so the only thing that matters is the negative 1 and the b. So the sign of this number is the same as the sign of negative b and similarly the sign of this number is  the sign of a times b because you know out of here the a squared is gone and the b to the fourth is gone and only an a and a b are left. So negative b and ab have to have the  same sign and so a is definitely negative. So our final answer is e.

Video Summary

The problem involves determining the sign of two expressions: \(-2a^4b^3c^2\) and \(3a^3b^5c^{-4}\), given that they have the same sign. The key insight is that even powers do not affect the sign, so these can be ignored. For the first expression, ignoring even powers, its sign reduces to \(-b\). For the second expression, simplifying under the same principle, its sign reduces to \(ab\). Thus, \(-b\) and \(ab\) must have the same sign, implying that \(a\) must be negative. Therefore, the correct answer is that \(a\) is definitely negative.

Keywords

expression sign

even powers

negative a

simplifying expressions

same sign