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Question

# Simplify each expression. Assume all variables are nonzero.$\left(\frac{r^2 s}{s^3}\right)^2$

Solution

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We have to simplify the expression:

\begin{aligned} \left(\frac{r^2s}{s^3}\right)^2&= \frac{\left(r^2s\right)^2}{\left(s^3\right)^2} & \text{(Power of quotient property)}\\ &= \frac{\left(r^2\right)^2s^2}{\left(s^3\right)^2} & \text{(Power of product property)}\\ &= \frac{r^{2 \cdot 2}s^2}{s^{3\cdot 2}} & \text{(Power of power property)}\\ &= \frac{r^{4}s^2}{s^{6}} \\ &= r^{4} s^{2-6} & \text{(Quotient of power property)} \\ &= r^{4} s^{-4} \\ &=\frac{r^{4}}{s^{4}} & \text{(Negative exponent property)}.\\ \end{aligned}

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