What does it mean to be simple

[GreenPrint]

What does it mean to be "simple"

Homework Statement

I never understood what it meant to be "simple"

For example in the ones below which one is more "simple" I was asked to simplify and didn't know were to stop as I am confused as to what it means to be "simple" as it's always defined very loosely and broadly, if somebody could give me a solid definition that I can apply to any problem when asked to "simplify" that would be great... like saying perform all the operations you can is wrong right? Because you could just completely expand everything and setting real numbers to imaginary numbers so that way there all in the same base ect... I'm just really confused what it means to be "simple"

-(14x + 21)/(x^2 + 3x) or -7/x - 7/(x+3)

(x^2 - 1)/(x^2 SQRT(x^2 + 3) + 2x^2) or 1/( SQRT(x^2 + 3) + 2 ) - 1//( x^2 SQRT(x^2 + 3) + 2x^2 )

Homework Equations

The Attempt at a Solution

 

[Mark44]

I don't know if you're going to get a hard and fast definition, although it should be fairly evident in context.

For example, if you are asked to simplify 2 + 3x -5 + 8x, you would combine like terms to get -3 + 11x. For the examples you gave, in the first one, the expression on the left is probably not simpler, but it is more useful for some things, such as finding x and y intercepts and asymptotes.

In the second example, the first expression is more compact, so I guess I would say that it's simpler.

 

[GreenPrint]

lol it's ok I'm unsure about it as well...
can somebody who knows for sure can answer which ones are more simpler and why?
see I don't think, "the most compact form" as a definition of "simple" is really correct you know?
But then again I don't think "perform all operations that you can" is really a good definition either...
which is why I still have no idea which forms are more "simple"

also note that I was specifically told to rationalize the numerator in the second problem and leave it in the denominator just in case you were wondering...

I hate doing that 1/SQRT(2) really is saying (SQRT(2)^(2^(-1))^-1... really stupid to do but that's not the point I'm trying to figuer out which is more simple in those problems...

-(14x + 21)/(x^2 + 3x) or -7/x - 7/(x+3)

I agree my instinct would be the second one because it has been split up into a fraction with only one entity in the numerator, -7, and one entity in the denominator, x, well I guess the whole thing is negative not the seven but whatever =O... but I'm not sure as I really have no idea what it means to be "simple" and don't know which expression is correct

(x^2 - 1)/(x^2 SQRT(x^2 + 3) + 2x^2) or 1/( SQRT(x^2 + 3) + 2 ) - 1//( x^2 SQRT(x^2 + 3) + 2x^2 )

sure the one on the left is more compact but does that make it more "simple" I'm stumped to =(

 

[GreenPrint]

do you think if I posted this in the calc section maybe somebody on there could help? Could I do that?

 

[Mark44]

What I'm saying is that there aren't any solid rules about simplifying expressions, other than the obvious example I gave.

In the past it was important to "simplify" expressions such as 1/sqrt(2), because there weren't any calculators to do the division for you, so it was much easier to calculate sqrt(2)/2 than it was to divide 1 by sqrt(2). So even though 1/sqrt(2) is exactly equal to sqrt(2)/2, the latter expression is simpler in terms being able to be calculated using only paper and pencil.

You probably won't see many "simplification"-type problems after elementary algebra, so IMO, it's not something that's worth dwelling on.

 

[GreenPrint]

ok thanks so what do you think I should put down for the answer?

 

[GreenPrint]

I would say the second one in both cases because you can't operate any furhter operations on them right?

 

[Pengwuino]

As has been stated, there are are no hard and fast rules about simplifying. In fact, an answer, no matter how well you simplify it, is still an answer. Of course, that isn't a license to never simplify because the simplified answers might make some important properties very obvious.

For example, in the first question, it's very obvious what the roots to that equation are using the first method. On the other hand, in the second equation, its painfully clearly something happens at x=0 and x = -3... although it's fairly obvious in the first method as well but i hope you get the point...