d) $$ \left(\frac{r^{3} s^{-1}}{s^{-2} r^{-2}}\right)^{-\frac{2}{3}} $$

Question

UpStudy AI · Accepted Answer

Given the expression
$$
\left(\frac{r^{3} s^{-1}}{s^{-2} r^{-2}}\right)^{-\frac{2}{3}},
$$
we simplify it step by step.

1. Combine the numerator and denominator by multiplying by the reciprocal:
   $$
   \frac{r^{3} s^{-1}}{s^{-2} r^{-2}} = r^{3} s^{-1} \cdot s^{2} r^{2}.
   $$

2. Simplify by combining the like bases:
   $$
   r^{3} \cdot r^{2} = r^{3+2} = r^5,
   $$
   $$
   s^{-1} \cdot s^{2} = s^{-1+2} = s.
   $$
   So, the expression inside the parentheses becomes:
   $$
   r^{5}s.
   $$

3. Raise the simplified expression to the power $$-\frac{2}{3}$$:
   $$
   \left(r^{5}s\right)^{-\frac{2}{3}} = r^{5\left(-\frac{2}{3}\right)} s^{-\frac{2}{3}} = r^{-\frac{10}{3}} s^{-\frac{2}{3}}.
   $$

Thus, the final simplified expression is
$$
\frac{1}{r^{\frac{10}{3}}s^{\frac{2}{3}}}.
$$
The simplified expression is $$ \frac{1}{r^{\frac{10}{3}}s^{\frac{2}{3}}} $$.

d) \( \left(\frac{r^{3} s^{-1}}{s^{-2} r^{-2}}\right)^{-\frac{2}{3}} \)

Upstudy AI Solution

Answer

Solution

Bonus Knowledge

Related Questions

Latest Algebra Questions