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Sometimes when I am doing my maths homework I sometimes get confused and then decide to make two mathematical statements and then plug some numbers in them to see if the results match. For example :

$$\frac{\sqrt{x}}x = \frac{1}{\sqrt{x}}$$

Is there any website/software that lets me type the left hand side and the right hand side of the above equation and then tells me if these two are equivalent statements or not(like telling me in the form of true and false) I searched but was unable to find any, so if you know it then the answer would be appreciated. Thnx in advance. Note: Since I am an Idiot and Newbie I was unable to find the relevant tags for the question.

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  • $\begingroup$ You could try putting it into WolframAlpha $\endgroup$ Jan 2 at 12:34
  • $\begingroup$ Also try the Boole command $\endgroup$ Jan 2 at 12:35
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    $\begingroup$ It may not be meaningful when $x=0$ $\endgroup$
    – Henry
    Jan 2 at 12:41
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    $\begingroup$ @Henry True, but in fact both expressions are invalid for $x=0$ , so even at this point , they are equivalent. $\endgroup$
    – Peter
    Jan 2 at 12:51
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    $\begingroup$ I would edit your 'statements' to 'expressions'; a statement is like a sentence and says something, while an expression is more like a noun and doesn't assert anything by itself. Your equation is one statement, and each side of the equals sign has one expression. "Are these two statements equivalent?" can be an interesting question in itself, but it isn't the one you're asking. $\endgroup$
    – dbmag9
    Jan 2 at 21:13

2 Answers 2

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Perhaps surprisingly, the answer is no in general. For a certain fairly natural class of expressions, it is undecidable whether a particular expression $E$ satisfies the equation $E = 0$. This is Richardson's theorem. Knowing whether $E1=E2$ is the same as knowing whether $E=E1-E2=0$.

Of course, in any particular case, you may well be able to find an argument as to why two expressions are equal. In your case you could obviously multiply top and bottom by $\sqrt{x}$. WolframAlpha will also help you.

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If you have a graphing utility (and Wolfram Alpha would work or a graphing calculator) you can graph both expressions. If they look the same then they probably are the same.

In the case of indefinite integrals, the two graphs should look the same but one might be shifted up from the other.

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