### Instruction: Find all solutions to the equation\[ \sqrt[4]{x} = \frac{12}{7 - \sqrt[4]{x}}.\].### Response: This is an equation involving a fourth root, so I want to get rid of it by isolating it on one side of the equation.
I can do this by multiplying both sides by \(7 - \sqrt[4]{x}\), which gives me\[ \sqrt[4]{x} (7 - \sqrt[4]{x}) = 12.\]
Now I can expand the left-hand side using the distributive property and get\[ 7\sqrt[4]{x} - (\sqrt[4]{x})^2 = 12.\]
I notice that this is a quadratic equation in disguise, where \(\sqrt[4]{x}\) is playing the role of the variable.
To make it look more familiar, I can let \(u = \sqrt[4]{x}\), so the equation becomes\[ 7u - u^2 = 12.\]
I can rearrange this into standard form and get\[ u^2 - 7u + 12 = 0.\]
Now I can factor this quadratic equation and get\[ (u - 4)(u - 3) = 0.\]
This means that either \(u - 4 = 0\) or \(u - 3 = 0\), so \(u = 4\) or \(u = 3\).
But \(u\) is not the original variable I want to solve for, it is just a substitution I made to simplify the equation.
So I need to go back and plug in \(u = \sqrt[4]{x}\) and get the values of \(x\).
If \(u = 4\), then \(\sqrt[4]{x} = 4\), which means that \(x\) must be the fourth power of \(4\), or \(4^4\).
If \(u = 3\), then \(\sqrt[4]{x} = 3\), which means that \(x\) must be the fourth power of \(3\), or \(3^4\).
So the solutions for \(x\) are \(4^4\) and \(3^4\), which are \(256\) and \(81\) respectively.