Question

Find the quotient.

$\frac{{\mathbf{x}}^{\mathbf{4}}}{{\mathbf{y}}^{\mathbf{2}}}\mathbf{÷}\frac{\mathbf{x}}{\mathbf{y}}$

Short answer

The final answer is $\frac{{\mathbf{x}}^{\mathbf{3}}}{\mathbf{y}}$.

Step-by-step solution

Step 1. Define a rational function.

A function of the form $y=\frac{p}{q}$is called a rational function, where $p$and$q$ are the polynomials and $q\ne 0$ .

Step 2. Define the quotient of two expressions ab and cd.

The quotient of two expressions $\frac{a}{b}$ and $\frac{c}{d}$, where $b\ne 0$ and $d\ne 0$, is given below.

$\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}\times \frac{d}{c}=\frac{ac}{bd}\mathrm{...........}\left(1\right)$

Step 3. Calculate the quotient x4y2÷xy.

Use $\left(1\right)$to find the quotient $\frac{x^4}{y^2}÷\frac{x}{y}$.

$\begin{array}{c}\frac{x^4}{y^2}÷\frac{x}{y}=\left(\frac{x^4}{y^2}\right)\times \left(\frac{y}{x}\right)\\ =\frac{\left(x^4\right)\left(y\right)}{\left(y^2\right)\left(x\right)}\\ =\frac{x^3}{y}\end{array}$

Therefore, the final answer is $\frac{x^3}{y}$.