enigma88 wrote:A general question
Which of the following is |a|^2/a equal to
1)a^2/|a|
2)|a/|a||*a
3) ||a|/a|*a

I'm guessing we're supposed to assume that a is not zero in this question.

Be aware of when you can eliminate absolute value bars. |a|^2 is always equal to a^2, because squaring a real number always makes it non-negative anyway, so the absolute value bars are superfluous. So you can rewrite |a|^2/a as a^2/a, which simplifies to just a, as long as a is not zero. So we're trying to figure out which expression is always equal to a.

1) a^2/|a|. First, notice that this expression must always be positive because it is a squared number divided by an absolute value. 'a' itself, however, does not have to be positive; it could be negative. This is enough to show that the two expressions are not always equal. To finish the analysis: Note that if a<0, |a|=-a, and when a>0, |a|=a. So This expression will simplify to -a for a<0, and a for a>0.

2) Note that a/|a| will always be either 1 or -1 for non-zero values of a. It will be -1 for a<0 and 1 for a>0. Thus, |a/|a|| will always be 1. So, this expression will always equal 1*a or just a.

3) Basically the same analysis as number 2. |a|/a will always be -1 or 1 for non-zero values of a, so the expression simplifies to just a.

Ans: 2 and 3