# Extraneous roots

## Main Question or Discussion Point

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:

$$\sqrt{x}-\sqrt[4]{x} -2=0$$
.
.
.
$$(x-16)(x-1)=0$$

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.

$$(x-16)(x-1)=0$$

$$x^{2}-17x+16=0$$

$$x^{2}-17x+16+25x=25x$$

$$x^{2}+8x+16=25x$$

$$(x+4)^{2}=25x$$

$$x+4=5\sqrt{x}$$

$$x+4-4\sqrt{x}=5\sqrt{x}-4\sqrt{x}$$

$$x+4-4\sqrt{x}=\sqrt{x}$$

$$(\sqrt{x}-2)^{2}=\sqrt{x}$$

$$(\sqrt{x}-2)^{2}=\sqrt{x}$$ equation A

$$\sqrt{x}-2=\sqrt[4]{x}$$ equation B

$$\sqrt{x}-\sqrt[4]{x}-2=0$$

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

$$((\sqrt{x}-2)^{2})^{\frac{1}{2}}$$

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?

$$x = 5, x^2= 25, x=5, -5$$. Exactly the same as that, but more disguised =] In this same one, when you squared it, you introduced the erroneous negative square root when only the positive root applies.