Question

if r and s are positive real numbers, which expression is equivalent to \\(\frac{r^{2/3}s^{1/2}}{rs}\\)
a \\(r^{-1}s^{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: Rewrite denominator exponents

Rewrite $r$ as $r^1$ and $s$ as $s^1$, so the expression becomes $\frac{r^{2/3}s^{1/2}}{r^1s^1}$

Step2: Apply exponent subtraction rule

For like bases, $\frac{x^a}{x^b}=x^{a-b}$.
For $r$: $r^{\frac{2}{3}-1}=r^{\frac{2}{3}-\frac{3}{3}}=r^{-\frac{1}{3}}$
For $s$: $s^{\frac{1}{2}-1}=s^{\frac{1}{2}-\frac{2}{2}}=s^{-\frac{1}{2}}$
Combine: $r^{-\frac{1}{3}}s^{-\frac{1}{2}}$

Step3: Rewrite negative exponents

Use $x^{-a}=\frac{1}{x^a}$, so $r^{-\frac{1}{3}}s^{-\frac{1}{2}}=\frac{1}{r^{1/3}s^{1/2}}$

Answer:

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