if (r) and (s) are positive real numbers, which expression is equivalent to (\frac{r^{2/3}s^{1/2}}{rs})?\na…

if (r) and (s) are positive real numbers, which expression is equivalent to (\frac{r^{2/3}s^{1/2}}{rs})?\na. (r^{3}s^{2})\nb. (r^{1/3}s^{1/2})\nc. (r^{2/3}s^{1/2})\nd. (\frac{1}{r^{2/3}s^{1/2}})\ne. (\frac{1}{r^{1/3}s^{1/2}})

if (r) and (s) are positive real numbers, which expression is equivalent to (\frac{r^{2/3}s^{1/2}}{rs})?\na. (r^{3}s^{2})\nb. (r^{1/3}s^{1/2})\nc. (r^{2/3}s^{1/2})\nd. (\frac{1}{r^{2/3}s^{1/2}})\ne. (\frac{1}{r^{1/3}s^{1/2}})

Answer

Explanation:

Step1: Use exponent - division rule

According to the rule $\frac{a^m}{a^n}=a^{m - n}$, for the given expression $\frac{r^{2/3}s^{1/2}}{rs}$, we can split it into two parts: $\frac{r^{2/3}}{r}$ and $\frac{s^{1/2}}{s}$.

Step2: Simplify the $r$ - part

For $\frac{r^{2/3}}{r}$, since $r = r^1$, then $\frac{r^{2/3}}{r^1}=r^{\frac{2}{3}-1}=r^{\frac{2 - 3}{3}}=r^{-1/3}$.

Step3: Simplify the $s$ - part

For $\frac{s^{1/2}}{s}$, since $s = s^1$, then $\frac{s^{1/2}}{s^1}=s^{\frac{1}{2}-1}=s^{\frac{1 - 2}{2}}=s^{-1/2}$.

Step4: Combine the results

The original expression $\frac{r^{2/3}s^{1/2}}{rs}=r^{-1/3}s^{-1/2}=\frac{1}{r^{1/3}s^{1/2}}$.

Answer:

E. $\frac{1}{r^{1/3}s^{1/2}}$