Question

divide.\\(
\frac{8u^{4}x^{5} - 15u^{7}x^{4} + 20u^{7}x}{-4u^{3}x^{2}}\\)
simplify your answer as much as possible.

Explanation:

Step1: Divide each term in the numerator by the denominator

We have the fraction \(\frac{8u^{4}x^{5}-15u^{7}x^{4}+20u^{7}x}{-4u^{3}x^{2}}\), which can be split into three separate fractions:
\(\frac{8u^{4}x^{5}}{-4u^{3}x^{2}}-\frac{15u^{7}x^{4}}{-4u^{3}x^{2}}+\frac{20u^{7}x}{-4u^{3}x^{2}}\)

Step2: Simplify each fraction using exponent rules (\(a^m\div a^n = a^{m - n}\))

• For the first fraction \(\frac{8u^{4}x^{5}}{-4u^{3}x^{2}}\):

The coefficient is \(\frac{8}{-4}=-2\), for \(u\) we have \(u^{4-3}=u^{1}=u\), and for \(x\) we have \(x^{5 - 2}=x^{3}\). So the first fraction simplifies to \(-2ux^{3}\).

• For the second fraction \(\frac{- 15u^{7}x^{4}}{-4u^{3}x^{2}}\):

The coefficient is \(\frac{-15}{-4}=\frac{15}{4}\), for \(u\) we have \(u^{7-3}=u^{4}\), and for \(x\) we have \(x^{4 - 2}=x^{2}\). So the second fraction simplifies to \(\frac{15}{4}u^{4}x^{2}\).

• For the third fraction \(\frac{20u^{7}x}{-4u^{3}x^{2}}\):

The coefficient is \(\frac{20}{-4}=-5\), for \(u\) we have \(u^{7-3}=u^{4}\), and for \(x\) we have \(x^{1-2}=x^{-1}=\frac{1}{x}\). So the third fraction simplifies to \(-5u^{4}\frac{1}{x}=-\frac{5u^{4}}{x}\).

Answer:

\(-2ux^{3}+\frac{15}{4}u^{4}x^{2}-\frac{5u^{4}}{x}\)