
#1
Apr308, 12:14 PM

P: 67

Sorry for the length of the post, the problem I've included is not difficult but I wanted to have an example to help illustrate my question.
solve: [tex]\sqrt{x}\sqrt[4]{x} 2=0 [/tex] . . . [tex](x16)(x1)=0[/tex] The roots are 16 and 1, however when one puts them back into the original equation, 1 is found to be extraneous leaving 16 as the only solution. My question is, why do extraneous roots arise? I attempted to answer the question myself by reversing the above process and putting 1 in for x at each step to see when the equation becomes "invalid" for the extraneous root. [tex](x16)(x1)=0[/tex] [tex]x^{2}17x+16=0[/tex] [tex]x^{2}17x+16+25x=25x[/tex] [tex]x^{2}+8x+16=25x[/tex] [tex](x+4)^{2}=25x[/tex] [tex]x+4=5\sqrt{x}[/tex] [tex]x+44\sqrt{x}=5\sqrt{x}4\sqrt{x}[/tex] [tex]x+44\sqrt{x}=\sqrt{x}[/tex] [tex](\sqrt{x}2)^{2}=\sqrt{x}[/tex] [tex](\sqrt{x}2)^{2}=\sqrt{x}[/tex] equation A [tex]\sqrt{x}2=\sqrt[4]{x}[/tex] equation B [tex]\sqrt{x}\sqrt[4]{x}2=0[/tex] Putting 1 in for x in equation A works but B does not. It seems that going from A to B creates the problem. When one takes the square root of equation A the left side becomes [tex]((\sqrt{x}2)^{2})^{\frac{1}{2}}[/tex] If I understood CompuChip's answer correctly to one of my previous posts, the inner to outer priority is not followed. If 1 is in for x, then 1 is the value in the first set of parenthesis and then 1 squared is 1, and then the square root is also 1. However if 1 is not in for x , since the roots are not known when one first goes through the problem, the squared to the 1/2 power gives what's in the parenthesis to the first power, which is just what's in the parenthesis. Then when 1 is in for x, we have 1 to the first power which is 1. The order of operations makes a difference for x=1 but does not for x=16. Is it true then, that extraneous roots arise because some mathematical operation is violated for that root? Thanks for any answers. 



#2
Apr308, 12:34 PM

P: 258

because equation B is not your original equation, you changed the power, and because of that you changed the roots



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