Answer
$x^{1/3}z^{5/6}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Use the definition of rational exponents and the laws of exponents to simplify the given expression, $
\sqrt[3]{xz}\cdot\sqrt[]{z}
.$
$\bf{\text{Solution Details:}}$
Using the definition of rational exponents which is given by $a^{\frac{m}{n}}=\sqrt[n]{a^m}=\left(\sqrt[n]{a}\right)^m,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
(xz)^{1/3}\cdot z^{1/2}
.\end{array}
Using the extended Power Rule of the laws of exponents which is given by $\left( x^my^n \right)^p=x^{mp}y^{np},$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
x^{1/3}\cdot z^{1/3}z^{1/2}
.\end{array}
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}
x^{\frac{1}{3}}z^{\frac{1}{3}+\frac{1}{2}}
.\end{array}
To simplify the expression $
\dfrac{1}{3}+\dfrac{1}{2}
,$ change the expressions to similar fractions (same denominator) by using the $LCD$. The $LCD$ of the denominators $
3
$ and $
2
$ is $
6
$ 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}
x^{\frac{1}{3}}z^{\frac{1}{3}\cdot\frac{2}{2}+\frac{1}{2}\cdot\frac{3}{3}}
\\\\=
x^{\frac{1}{3}}z^{\frac{2}{6}+\frac{3}{6}}
\\\\=
x^{\frac{1}{3}}z^{\frac{5}{6}}
\\\\=
x^{1/3}z^{5/6}
.\end{array}
