b) \( \sqrt[6]{x+16}=x-4 \)

To solve the equation \( \sqrt[6]{x+16} = x - 4 \), we first raise both sides to the 6th power to eliminate the radical. This gives us: \[ x + 16 = (x - 4)^6 \] Now, expanding \( (x - 4)^6 \) can be quite complex, but it's essential to recognize that this equation will generally yield multiple solutions due to the nature of polynomial equations. Next, we can start by simplifying the equation to look for possible values of \( x \) that satisfy it. To find the solutions more easily, we can try substituting some rational numbers that could satisfy the equation. For instance, testing \( x = 4 \): \[ \sqrt[6]{4 + 16} = 4 - 4 \Rightarrow \sqrt[6]{20} = 0 \quad \text{(not a solution)} \] Instead, let's try \( x = 5 \): \[ \sqrt[6]{5 + 16} = 5 - 4 \Rightarrow \sqrt[6]{21} = 1 \quad \text{(not a solution)} \] Continuing this testing or even graphical methods could lead to valid values of \( x \). Once we have a polynomial equation established, we can also solve it using numerical methods or graphing tools for a more accurate approximation. Always remember to check for extraneous solutions since raising both sides of an equation can introduce them!