The quotient of the square of a number minus the cubed root of another number and the sum of those two numbers is nine. How can this relationship best be stated algebraically? A \( \left(x^{2}-\sqrt[3]{y}\right)-(x+y)=0 \)

Let's break it down! We want to express the idea that the quotient of a certain expression equals nine. Specifically, you can say that the square of a number \(x\) minus the cube root of another number \(y\) divided by the sum of those two numbers (i.e., \(x + y\)) is equal to nine. Algebraically, this can be represented as: \[ \frac{x^{2} - \sqrt[3]{y}}{x + y} = 9. \] Now for a common mistake: when dealing with quotients, people sometimes forget to carry the denominator through the algebraic manipulation. Make sure to clearly define your expressions before dividing! It’s easy to mix signs or forget to include both terms in your final verification, leading to headaches later on in problem-solving.