Properties of Exponents Issue w/ HP Prime

[Joe7248]

I input x^(-1/2) into the HP Prime and expect when simplifying this problem for the HP Prime to give this answer 1/x^(1/2) but instead I receive an answer of x^(1/2)/x. I have used the TI-89 Titanium, Casio Classpad 400, and Wolfram Alpha confirming that simplifying x^(-1/2) the answer should be 1/x^(1/2) and not x^(1/2)/x. I have already set under the CAS Settings for Simplify to Maximum.

[parisse]

Both answers are correct.

[ww63]

Hello,

shure both answers are correct, but i think you agree 1/x^(1/2) is simplier than x^(1/2)/x

[parisse]

No, I prefer to have square roots in the numerator. Same as for 1/(1+i) and (1-i)/2=1/2-i/2

[Joe7248]

I agree with you ww63. When I set the simplify setting under Maximum, the answer I expect to see is 1/x^(1/2) and x^(1/2)/x can be a response under the setting Minimum. It's not convenient either when seeking the derivative to a problem using the quotient rule and the variable x has a negative exponent in fraction form. I would appreciate a software programmer from HP to change this annoying response from the calculator by giving an answer like those of other established calculators.

[Helge Gabert]

I agree with Bernard Parisse - - leave simplify() the way it is if it consistently leaves square roots in the numerator.

But if one would like to obtain 1/sqrt(x), square root in the denominator, just use collect():

collect(x^(-1/2) returns 1/sqrt(x), and also
collect(sqrt(x)/x) returns 1/sqrt(x).

[Joe7248]

I might be doing it wrong forgive me but I used collect and entered x^(-1/2) and the calculator still returned x^(1/2)/x. Here's another concern with this issue... I used the TI-89 Titanium and Wolfram Alpha to get the derivative of: (4 sqrt(x))/(x^2-2)) and the answer equaled -(2 (3 x^2+2))/(sqrt(x) (x^2-2)^2) on both which was great. If needed here is the Wolfram Alpha Language Plain Text Input: D[(4 Sqrt[x])/(x^2-2), x] to see the problem in textbook format. However, the HP Prime provides this as an answer to (4x^(1/2))/(x^2-2)) using the differentiate (diff) option & Maximum Simplify setting (-6x^2*sqrt(x)-4*sqrt(x))/(x^5-4*x^3+4*x) it can also be copied and pasted to Wolfram Alpha to see its textbook format and understand my problem. HP Programmers please correct this its my belief that the answers which have appeared on my HP Prime Calculator, should only appear if Simplify is set to Minimum because the answers can be simplified further. No Excuses! Your answers should not end halfway simplified under Maximum settings. I have faith in your abilities to address this issue and satisfy my concern.

[John P]

(12-12-2014 03:06 AM)Joe7248 Wrote:  I agree with you ww63. When I set the simplify setting under Maximum, the answer I expect to see is 1/x^(1/2) and x^(1/2)/x can be a response under the setting Minimum. It's not convenient either when seeking the derivative to a problem using the quotient rule and the variable x has a negative exponent in fraction form. I would appreciate a software programmer from HP to change this annoying response from the calculator by giving an answer like those of other established calculators.

I remember fom my algebra cources that leaving roots in the denominator was always a bad idea. Expression was considered simplified when there was no roots in the denominator.

[Joe Horn]

(12-12-2014 06:08 AM)John P Wrote:  I remember fom my algebra cources that leaving roots in the denominator was always a bad idea. Expression was considered simplified when there was no roots in the denominator.

A succinct explanation of the history of "rationalizing denominators", with the pros and cons of still teaching it (and doing it), is presented by "Dr. Math" here: http://mathforum.org/library/drmath/view/52663.html

[Han]

(12-12-2014 06:04 AM)Joe7248 Wrote:  I might be doing it wrong forgive me but I used collect and entered x^(-1/2) and the calculator still returned x^(1/2)/x.

Have you updated to the most recent firmware? What are your CAS settings? Did you make sure to put CAS settings on Minimum? Also, my calculator uses the \( \sqrt{} \) symbol whereas you are suggesting it returns a fractional exponent.

Quote:Here's another concern with this issue... I used the TI-89 Titanium and Wolfram Alpha to get the derivative of: (4 sqrt(x))/(x^2-2)) and the answer equaled -(2 (3 x^2+2))/(sqrt(x) (x^2-2)^2) on both which was great. If needed here is the Wolfram Alpha Language Plain Text Input: D[(4 Sqrt[x])/(x^2-2), x] to see the problem in textbook format. However, the HP Prime provides this as an answer to (4x^(1/2))/(x^2-2)) using the differentiate (diff) option & Maximum Simplify setting (-6x^2*sqrt(x)-4*sqrt(x))/(x^5-4*x^3+4*x) it can also be copied and pasted to Wolfram Alpha to see its textbook format and understand my problem. HP Programmers please correct this its my belief that the answers which have appeared on my HP Prime Calculator, should only appear if Simplify is set to Minimum because the answers can be simplified further. No Excuses! Your answers should not end halfway simplified under Maximum settings. I have faith in your abilities to address this issue and satisfy my concern.

Are you suggesting that \[ \frac{-6x^2\sqrt{x}-4\sqrt{x}}{x^5-4x^3+4x} \] can be simplified more? In what sense of "simplify"? I don't see how you can simplify this more except to possibly multiply the \( x^2 \sqrt{x} \) into a single power of x. The expression
\[ -\frac{2 (3 x^2+2)}{\sqrt{x} (x^2-2)^2} \]
is not simplified because there are multiplication operations that have not been evaluated. I'm only being half-serious here. My point is that "to simplify" in mathematics does not always mean the same thing to everyone.

Using minimum simplification and collect on the result gives the same fraction as the ti89 and wolfram alpha on my machine.


   


Edit: Consider \( (x-b)^2 - (a+x)^2 = -(a+b) (2x+a-b) = -2(a+b)x + (b^2-a^2) \). All three forms are arguably in simplest form depending on what you plan to do with the expression. The first expression collects everything in terms of a and b separately, the second form is nice for finding the zeros of the expression, and the last form is useful if you want to simplify into a polynomial.

[John P]

(12-12-2014 10:37 AM)Joe Horn Wrote:  

(12-12-2014 06:08 AM)John P Wrote:  I remember fom my algebra cources that leaving roots in the denominator was always a bad idea. Expression was considered simplified when there was no roots in the denominator.

A succinct explanation of the history of "rationalizing denominators", with the pros and cons of still teaching it (and doing it), is presented by "Dr. Math" here: http://mathforum.org/library/drmath/view/52663.html

Of course, and as always: the Golden Rule of Life applies: "everything in moderation". Sometimes with fraktions or with complex numbers you must multiply by the complemet of the denominator and it all depends 'where you want to go' so you must know what you are doing. Even with computers and calculators you should know 'where you want to go' and whot you are doing and not only rely on what the machine spits on you.

[Helge Gabert]

(12-12-2014 06:04 AM)Joe7248 Wrote:  I might be doing it wrong forgive me but I used collect and entered x^(-1/2) and the calculator still returned x^(1/2)/x.

For collect() to work as described, please set simplify in the CAS settings to "Minimum." Sorry, I should have indicated that right away.

I have found that this CAS setting ("Minimum") works quite well for the majority of my calculations and manipulations of expressions.

[Joe7248]

I appreciate everyone's comments and would like to say thank you for your help in finding the answer I was looking for. On the other hand, I would like to see my numerator and denominator factored under Maximum setting. Yet, I understand others point of view that its relative to what you are seeking and how you plan to use your answer. I personally prefer, the setup from the other calculators not to mean that anyone else preference is not valuable. I still call on HP Prime to consider what i have been wanting. I do not want to have to constantly seek Collect for the answer I'm looking for under Minimum setting, its annoying for me. Please place an option in the background that can be programmed to help me avoid that. P.S. Please also put an option for a user like myself who does not want to see x^(-1/2)=(x^(1/2))/(x) but x^(-1/2)=1/x^(1/2)

[John P]

(12-12-2014 06:39 PM)Joe7248 Wrote:  I appreciate everyone's comments and would like to say thank you for your help in finding the answer I was looking for. On the other hand, I would like to see my numerator and denominator factored under Maximum setting. Yet, I understand others point of view that its relative to what you are seeking and how you plan to use your answer. I personally prefer, the setup from the other calculators not to mean that anyone else preference is not valuable. I still call on HP Prime to consider what i have been wanting. I do not want to have to constantly seek Collect for the answer I'm looking for under Minimum setting, its annoying for me. Please place an option in the background that can be programmed to help me avoid that. P.S. Please also put an option for a user like myself who does not want to see x^(-1/2)=(x^(1/2))/(x) but x^(-1/2)=1/x^(1/2)

Prime is programmable so write your own program to do what you want and you set for life.

[Joe7248]

Thank you for the advice. John P you are right in Algebra it's taught to rationalize the denominator but once Calculus comes around that rule is not forced on students. Just like in Algebra courses improper fractions are accepted as answers yet, in Elementary school improper fractions were frowned upon. That's why on Wolfram Alpha and Texas Instruments CAS Calculators and Casio Classpad the denominator is allowed to have a radical when seeking a derivative. Plus, I'll have to brush up on my programming skills for this calculator. I should have done my research before purchasing this calculator. I love the hardware for this calculator though. HP could earn a customer for life if they'd look into my complaints and resolve my issue with a sensible option in the settings menu.

[parisse]

This is not just a small option. simplify calls normal, and the main algorithm of normal require that all square roots are in the numerator.
You have several commands to rewrite an expression: for example ratnormal does not simplify square roots (they are replaced by a fake identifier and rational normalization is called).
As Han wrote, there is no universal simplification command, it depends what you want to do after, that's why you have several rewrite commands on the HP CAS, it's the user job to choose the appropriate one. TI offers much less choice, if their auto-simplification does not return what you expect, it's much more difficult to get what you want.
Xcas lets you choose the simplification command that is automatically applied, perhaps the HP could offer the same choice instead of none/minimum/maximum, minimum corresponds to regroup, maximum corresponds to simplify. I do not recommend simplify, because it's much more intrusive and might require time, regroup (or none) is better.

[Joe7248]

Parisse your explanation was very helpful and gave me a better understanding of what the calculator wishes to accomplish. I'll have to dig deep into the manual and consider my options when getting an answer that will be appropriate. I love the brotherhood here at MoHPC it's great to know help is only post away.