Question

We were never taught this in class, and I understand it, kind of... but i'm just confused with everything.

Answer 1

$\bf \cfrac{x^3y}{xy^5}\cdot \cfrac{x^2y^9}{x^8}\implies \cfrac{x^3x^2y^1y^9}{x^1x^8y^5} \\ \quad \\ \textit{now, using exponent rules, same base, different exponent} \\ \quad \\ \cfrac{x^{3+2}y^{1+9}}{x^{1+8}y^5}\implies \cfrac{x^5y^{10}}{x^9y^5}\\ ----------------------------$$\bf \textit{now, keep in mind that} \\ \quad \\ a^{-{ n}} \implies \cfrac{1}{a^{ n}}\qquad \qquad \cfrac{1}{a^{ n}}\implies a^{-{ n}} \\ \quad \\ % negative exponential denominator a^{{ n}} \implies \cfrac{1}{a^{- n}} \qquad \qquad \cfrac{1}{a^{- n}}\implies \cfrac{1}{\frac{1}{a^{ n}}}\implies a^{{ n}} \\ \quad \\ thus\\ ----------------------------$$\bf \cfrac{x^5y^{10}}{x^9y^5}\implies \cfrac{x^5y^{10}}{1}\cdot \cfrac{1}{x^9}\cdot \cfrac{1}{y^5}\implies x^5y^{10}\cdot x^{\boxed{-9}}\cdot y^{\boxed{-5}} \\ \quad \\ x^5x^{-9}y^{10}y^{-5}\implies x^{5-9}y^{10-5}\implies x^{-4}y^5 \\ \quad \\ \cfrac{1}{x^{\boxed{4}}}\cdot \cfrac{y^5}{1}\implies \cfrac{y^5}{x^4}$