If $$ r $$ and $$ s $$ are positive real numbers, which expression is equivalent to $$ \frac{r^{2 / 3} s^{1 / 2}}{r s} ? $$ A. $$ r^{3} s^{2} $$ B. $$ r^{1 / 3} s^{1 / 2} $$ C. $$ r^{2 / 3} s^{1 / 2} $$ D. $$ \frac{1}{r^{2 / 3} s^{1 / 2}} $$ E. $$ \frac{1}{r^{1 / 3} s^{1 / 2}} $$

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UpStudy AI · Accepted Answer

To simplify the expression $$\frac{r^{2/3} s^{1/2}}{r s}$$, follow these steps:

1. **Express the Denominator with Exponents:**
   
   The denominator $$r s$$ can be written as $$r^1 s^1$$.

2. **Apply the Law of Exponents:**
   
   When dividing like bases, subtract the exponents:
   
   $$
   \frac{r^{2/3} s^{1/2}}{r^1 s^1} = r^{2/3 - 1} s^{1/2 - 1} = r^{-1/3} s^{-1/2}
   $$

3. **Rewrite with Positive Exponents:**
   
   Negative exponents indicate the reciprocal:
   
   $$
   r^{-1/3} s^{-1/2} = \frac{1}{r^{1/3} s^{1/2}}
   $$

Thus, the expression simplifies to:

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

**Answer:**  
$$\boxed{\dfrac{1}{\,r^{1/3}\, s^{1/2}\,}}$$
The expression simplifies to $$ \frac{1}{r^{1/3} s^{1/2}} $$.

If \( r \) and \( s \) are positive real numbers, which expression is equivalent to \( \frac{r^{2 / 3} s^{1 / 2}}{r s} ? \) A. \( r^{3} s^{2} \) B. \( r^{1 / 3} s^{1 / 2} \) C. \( r^{2 / 3} s^{1 / 2} \) D. \( \frac{1}{r^{2 / 3} s^{1 / 2}} \) E. \( \frac{1}{r^{1 / 3} s^{1 / 2}} \)

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