First, let's isolate [tex]x[/tex] on one side of the equation. To do this, we can subtract 1 from both sides of the equation:
[tex]x^{\frac{1}{4}} = -1[/tex]
Now, we are going to want to get x to a power of 1, since that is what we are looking for after all. To do this, we can set both sides of the equation to the exponent of 4:
[tex](x^{\frac{1}{4}})^4 = (-1)^4[/tex]
[tex]x = 1[/tex]
We have found x = 1. However, let's check this answer to make sure that it is not an extraneous solution:
[tex]1^{\frac{1}{4}} \stackrel{?}{=} -1[/tex]
[tex]1^{\frac{1}{4}} = 1 \neq -1[/tex]
When we substitute x = 1 into the original equation, we get 1, which is not -1. Thus x = 1 is an extraneous solution. Since there are no other values that we found, the equation has no solutions.
