Question

In the following exercises,simplify

$$\begin{gathered} \left(a\right)\frac{r^{{\displaystyle \frac{5}{2}}}.r^{-{\displaystyle \frac{1}{2}}}}{r^{-{\displaystyle \frac{3}{2}}}} \\ \left(b\right)\left(\frac{36s^{{\displaystyle \frac{1}{5}}}{t}^{-{\displaystyle \frac{3}{2}}}}{s^{-{\displaystyle \frac{9}{5}}}.t^{{\displaystyle \frac{1}{2}}}}\right) \end{gathered}$$

Short answer

$$\begin{gathered} \left(a\right)r^{\frac{7}{2}} \\ \left(b\right)36s^2{t}^{-2} \end{gathered}$$

Step-by-step solution

Part (a) Step 1.Given information

Consider the expression $\frac{r^{{\displaystyle \frac{5}{2}}}·r^{-{\displaystyle \frac{1}{2}}}}{r^{-{\displaystyle \frac{3}{2}}}}$

To simplify the expression, by using product and quotient property that is

$a^m·a^n=a^{m+n}and\frac{a^m}{a^n}=a^{m-n}$

Part (a) Step 2.Simplify the expression

$$\begin{gathered} \frac{r^{{\displaystyle \frac{5}{2}}}·r^{-{\displaystyle \frac{1}{2}}}}{r^{-{\displaystyle \frac{3}{2}}}}=\frac{r^{{\displaystyle \frac{5}{2}}-{\displaystyle \frac{1}{2}}}}{r^{-{\displaystyle \frac{3}{2}}}}[\sin ce;a^m·a^n=a^{m+n]} \\ =\frac{r^2}{r^{-{\displaystyle \frac{3}{2}}}}[Addtheexponent] \\ =r^{2-(-\frac{3}{2})}[Since;a^{\frac{m}{n}}=a^{m-n}] \\ =r^{2+\frac{3}{2}}[Simplifytheexponent] \\ =r^{\frac{7}{2}}[Simplify] \\ Therefore,theansweris{r}^{\frac{7}{2}} \end{gathered}$$

Part (b) Step 1.Given information

Consider the expression $\left(\frac{36s^{{\displaystyle \frac{1}{5}}}{t}^{-{\displaystyle \frac{3}{2}}}}{s^{-{\displaystyle \frac{9}{5}}}·t^{{\displaystyle \frac{1}{2}}}}\right)$

To simplify the expression, by using product and quotient property that is,

$a^m·a^n=a^{m+n}and\frac{a^m}{a^n}=a^{m-n}$

Part (b) Step 2.Simplify the expression

$$\begin{gathered} \left(\frac{36s^{{\displaystyle \frac{1}{5}}}{t}^{-{\displaystyle \frac{3}{2}}}}{s^{-{\displaystyle \frac{9}{5}}}·t^{{\displaystyle \frac{1}{2}}}}\right)=\left(36\right)·\left(s^{\frac{1}{5}-(-\frac{9}{5})}\right)·\left(t^{-\frac{3}{2}-\frac{1}{2}}\right)[Since\frac{a^m}{a^n}=a^{m-n}] \\ =\left(6^2\right)·\left(s^{\frac{1}{5}-\left(-\frac{9}{5}\right)}\right)·\left(t^{-\frac{3}{2}-\frac{1}{2}}\right)[\sin ce;6^2=36] \\ =\left(6^2\right)·\left(s^2\right)·\left(t^{-2}\right)[substracttheexponent] \\ ={\left(6\right)}^2·{\left(s\right)}^2·{\left(t\right)}^{-2}[\sin ce;{\left(a^m\right)}^n=a^{mn}] \\ =36s^2{t}^{-2}[multiplaytheexponent] \\ Therefore,theansweris36s^2{t}^{-2} \end{gathered}$$