Question

divide.
\\((-37u^{4}z^{6} + 24u^{7}z^{6}) \div (-4u^{5}z^{4})\\)
simplify your answer as much as possible.

Explanation:

Step1: Split the division into two terms

We can split the numerator into two separate fractions by the rule \((a + b)\div c=\frac{a}{c}+\frac{b}{c}\). So we have:
\(\frac{-37u^{4}z^{6}}{-4u^{5}z^{4}}+\frac{24u^{7}z^{6}}{-4u^{5}z^{4}}\)

Step2: Simplify the first fraction

For the first fraction \(\frac{-37u^{4}z^{6}}{-4u^{5}z^{4}}\), we use the rule of exponents \(\frac{a^{m}}{a^{n}} = a^{m - n}\) and the rule of dividing coefficients. The coefficients \(\frac{- 37}{-4}=\frac{37}{4}\), for \(u\) terms \(\frac{u^{4}}{u^{5}}=u^{4 - 5}=u^{-1}=\frac{1}{u}\), for \(z\) terms \(\frac{z^{6}}{z^{4}}=z^{6-4}=z^{2}\). So the first fraction simplifies to \(\frac{37z^{2}}{4u}\)

Step3: Simplify the second fraction

For the second fraction \(\frac{24u^{7}z^{6}}{-4u^{5}z^{4}}\), the coefficients \(\frac{24}{-4}=- 6\), for \(u\) terms \(\frac{u^{7}}{u^{5}}=u^{7 - 5}=u^{2}\), for \(z\) terms \(\frac{z^{6}}{z^{4}}=z^{6-4}=z^{2}\). So the second fraction simplifies to \(-6u^{2}z^{2}\)

Step4: Combine the two simplified fractions

Combining the two results from step 2 and step 3, we get \(\frac{37z^{2}}{4u}-6u^{2}z^{2}\)

Answer:

\(\frac{37z^{2}}{4u}-6u^{2}z^{2}\)