What is the prime factorization of the monomial $18x^{2}y$?

Perform the division and simplify the result to lowest terms: $\frac{x^2-1}{2x}÷\frac{x-1}{6}$

Simplify the expression fully and state all domain exclusions: $\frac{y^2-4}{y^2+2y}$

Simplify by canceling common factors: $\frac{14x}{42x^2}$

Simplify and state excluded values: $\frac{x^2-5x+6}{x^2-4}$

A pharmacist combines two liquid solutions: $\frac{4t+2}{10t}$ liters and $\frac{t-3}{10t}$ liters into one container. Write the simplified expression for the total volume and state any $t$-values that are not allowed.

The phrase "$y$ varies directly as $x$" in a word problem signals which mathematical model should be used?

Given that $y$ varies directly as $x$ and $y=12$ when $x=4$, find the equation relating $x$ and $y$.

A gardener finds that the required fertilizer mass $y$, in units of $\text{kg}$, will vary directly with the garden area $A$, in units of $\text{m}{}^2$, and will vary inversely with the fertilizer concentration $c$, in units of $\frac{\text{kg}}{{\text{m}}^2}$ (for the recommended strength), such that $y=\frac{kA}{c}$. If the gardener uses 50 $\text{kg}$ of fertilizer for a 200 ${\text{m}}^2$ plot with a concentration of $c=2$, determine how many kilograms will be needed for a 300 ${\text{m}}^2$ plot at concentration of $c=3$.