Integers problem

Question

When a,b,c are real numbers and ab²c^4 > 0 which of the following must be greater than 0?

 

a) ac²

b )ac

c) ab

d)abc

E)bc

 

Any help will be appreciated.THANKYOU!!

Answer 1

First, remember that multiplying two positive numbers or two negative numbers together will give you a positive number. Multiplying a positive number and a negative number together will give you a negative number.

 

As a consequence, any even power of a number (x², x⁴, x⁶, etc.) is a positive number.

 

You're told that ab²c⁴ > 0. In other words, it's positive. b² and c⁴ are guaranteed to be positive quantities no matter what b and c are, since they're even powers. That means that b²c⁴ is also positive. So, for the whole thing to be positive, a must be positive as well.

 

We therefore know that a is positive, b is either negative or positive, and c is either negative or positive.

 

Let's look at our answer choices.

 

(a) ac²

 

We know a is positive, and c² is an even power, so it's positive as well. Therefore, ac² must be greater than 0. This answer choice is correct. Let's check the others...

 

(b) ac

 

We know a is positive, but if c is negative, then ac is negative as well.

 

(c) ab

 

We know a is positive, but if b is negative, then ab is negative as well.

 

(d) abc

 

We know a is positive, but if bc is negative, then abc is negatve as well. This would happen if either b or c (but not both) were negative.

 

(e) bc

 

If either b or c (but not both) is negative, then bc will be negative as well.