Question

divide.\frac{-8y^{7}x + 7y^{7}x^{4}}{-2y^{5}x^{2}}\
simplify your answer as much as possible.

Explanation:

Step1: Split the fraction

We can split the given fraction into two separate fractions:
$$\frac{-8y^{7}x + 7y^{7}x^{4}}{-2y^{5}x^{2}}=\frac{-8y^{7}x}{-2y^{5}x^{2}}+\frac{7y^{7}x^{4}}{-2y^{5}x^{2}}$$

Step2: Simplify the first fraction

For the first fraction $\frac{-8y^{7}x}{-2y^{5}x^{2}}$, we divide the coefficients and use the rules of exponents for variables. The coefficient: $\frac{-8}{-2} = 4$. For $y$: $y^{7-5}=y^{2}$. For $x$: $x^{1 - 2}=x^{-1}=\frac{1}{x}$. So the first fraction simplifies to $4y^{2}\cdot\frac{1}{x}=\frac{4y^{2}}{x}$.

Step3: Simplify the second fraction

For the second fraction $\frac{7y^{7}x^{4}}{-2y^{5}x^{2}}$, the coefficient: $\frac{7}{-2}=-\frac{7}{2}$. For $y$: $y^{7 - 5}=y^{2}$. For $x$: $x^{4-2}=x^{2}$. So the second fraction simplifies to $-\frac{7}{2}y^{2}x^{2}$.

Step4: Combine the two simplified fractions

Now we combine the two results from Step 2 and Step 3:
$$\frac{4y^{2}}{x}-\frac{7}{2}x^{2}y^{2}$$

Answer:

$\frac{4y^{2}}{x}-\frac{7}{2}x^{2}y^{2}$