Answer
$ \displaystyle \frac{3k^{3/2}}{8}$
Work Step by Step
$\displaystyle \frac{3k^{2}\cdot(4k^{-3})^{-1}}{4^{1/2}\cdot k^{7/2}} \qquad$ ....... use $(ab)^{m}=a^{m} b^{m}$
$=\displaystyle \frac{3k^{2}\cdot 4^{-1}(k^{-3})^{-1}}{4^{1/2}\cdot k^{7/2}}\qquad$
....... use $(a^{m})^{n}=a^{mn}$, group like terms
$=\displaystyle \frac{3\cdot 4^{-1}}{4^{1/2}}\cdot\frac{k^{2}\cdot k^{3}}{k^{7/2}} \qquad$ ....... use $a^{m}\cdot a^{n}=a^{m+n}$
$=3\displaystyle \cdot\frac{4^{-1}}{4^{1/2}}\cdot\frac{k^{5}}{k^{7/2}}\qquad$ ....... use $\displaystyle \frac{a^{m}}{a^{n}}=a^{m-n}$
$=3(4^{-1-1/2})(k^{5-7/2)}$
$=3\cdot 4^{-3/2}\cdot k^{3/2}\qquad$ .......recognize: $4=2^{2}$
$=3(2^{2})^{-3/2}\cdot k^{3/2}\qquad$ ....... use $(a^{m})^{n}=a^{mn}$
$=3\cdot 2^{-3}\cdot k^{3/2}\qquad$ ....... use $a^{-n}=\displaystyle \frac{1}{a^{n}}=(\frac{1}{a})^{n}$
$= \displaystyle \frac{3k^{3/2}}{8}$
