## Intermediate Algebra (12th Edition)

$x^{1/3}z^{5/6}$
$\bf{\text{Solution Outline:}}$ 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}