Answer
$z^{3/2}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Use the laws of exponents to simplify the given expression, $
\dfrac{z^{3/4}}{z^{5/4}\cdot z^{-2}}
.$
$\bf{\text{Solution Details:}}$
Using the Product Rule of the laws of exponents which is given by $x^m\cdot x^n=x^{m+n},$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{z^{3/4}}{z^{\frac{5}{4}+(-2)}}
\\\\=
\dfrac{z^{3/4}}{z^{\frac{5}{4}-2}}
.\end{array}
Using the Quotient Rule of the laws of exponents which states that $\dfrac{x^m}{x^n}=x^{m-n},$ the expression above simplifies to
\begin{array}{l}\require{cancel}
z^{\frac{3}{4}-\left( \frac{5}{4}-2 \right)}
\\\\=
z^{\frac{3}{4}-\frac{5}{4}+2}
\\\\=
z^{-\frac{2}{4}+2}
\\\\=
z^{-\frac{1}{2}+2}
.\end{array}
To simplify the expression $
-\dfrac{1}{2}+2
,$ change the expressions to similar fractions (same denominator) by using the $LCD$. The $LCD$ of the denominators $
2
$ and $
1
$ is $
2
$ since it is the lowest number that can be exactly divided by the denominators. Multiplying the terms by an expression equal to $1$ that will make the denominator equal to the $LCD$ results to
\begin{array}{l}\require{cancel}
z^{-\frac{1}{2}+2\cdot\frac{2}{2}}
\\\\=
z^{-\frac{1}{2}+\frac{4}{2}}
\\\\=
z^{\frac{3}{2}}
\\\\=
z^{3/2}
.\end{array}
