Question

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

a. \\(r^3s^2\\)

b. \\(r^{1/3}s^{1/2}\\)

c. \\(r^{2/3}s^{1/2}\\)

d. \\(\frac{1}{r^{2/3}s^{1/2}}\\)

e. \\(\frac{1}{r^{1/3}s^{1/2}}\\)

Explanation:

Step1: Simplify the \( r \)-terms

Using the exponent rule \( \frac{a^m}{a^n}=a^{m - n} \), for the \( r \)-terms: \( \frac{r^{2/3}}{r^1}=r^{\frac{2}{3}-1} \). Calculate \( \frac{2}{3}-1=\frac{2}{3}-\frac{3}{3}=-\frac{1}{3} \), so the \( r \)-term becomes \( r^{-1/3} \).

Step2: Simplify the \( s \)-terms

For the \( s \)-terms: \( \frac{s^{1/2}}{s^1}=s^{\frac{1}{2}-1} \). Calculate \( \frac{1}{2}-1=\frac{1}{2}-\frac{2}{2}=-\frac{1}{2} \), so the \( s \)-term becomes \( s^{-1/2} \).

Step3: Rewrite negative exponents

Using the rule \( a^{-n}=\frac{1}{a^n} \), \( r^{-1/3}s^{-1/2}=\frac{1}{r^{1/3}s^{1/2}} \).

Answer:

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