Question

simplify. express your answer using positive exponents.
\frac{8rs^{4}}{r^{5}s^{2}}

Explanation:

Step1: Apply quotient - rule of exponents for r and s

For the variable \(r\), we have \(\frac{r}{r^{5}}=r^{1 - 5}\) and for \(s\), \(\frac{s^{4}}{s^{2}}=s^{4 - 2}\). The original expression \(\frac{8rs^{4}}{r^{5}s^{2}}\) can be written as \(8\times\frac{r}{r^{5}}\times\frac{s^{4}}{s^{2}}\).

Step2: Simplify the exponents of r and s

\(r^{1 - 5}=r^{-4}\) and \(s^{4 - 2}=s^{2}\). So the expression becomes \(8r^{-4}s^{2}\).

Step3: Rewrite with positive exponents

Using the rule \(a^{-n}=\frac{1}{a^{n}}\), \(r^{-4}=\frac{1}{r^{4}}\). So the simplified expression is \(\frac{8s^{2}}{r^{4}}\).

Answer:

\(\frac{8s^{2}}{r^{4}}\)