### Instruction: Find all solutions to the equation\[ \sqrt[4]{x} = \frac{12}{7 - \sqrt[4]{x}}.\].### Response: This equation looks tricky because it has a fourth root in it, but I notice that it has a similar structure to a quadratic equation: something equals a constant divided by another constant minus that same something.
I wonder if I can use a substitution to make this equation look more familiar.
Let me try letting\[ y = \sqrt[4]{x},\] so that the equation becomes\[ y = \frac{12}{7 - y}.\]
This looks much more manageable.
Now I can cross-multiply and get\[ 7y - y^2 = 12.\]
This is a quadratic equation, so I can rearrange it into standard form and try to factor or use the quadratic formula.
I get\[ y^2 - 7y + 12 = 0.\]
I notice that this is easily factorable, since\[ y^2 - 7y + 12 = (y - 3)(y - 4).\]
So this means that\[ y = 3 \quad \text{or} \quad y = 4.\]
But remember, y was not the original variable we wanted to solve for.
We have to go back to our substitution and plug in these values of y to get the values of x. So we have\[ x = y^4 = 3^4 \quad \text{or} \quad x = y^4 = 4^4.\]