## Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

$x\sqrt[4]{x}$
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 given expression is equivalent to \begin{array}{l}\require{cancel} \sqrt[4]{x^3}\sqrt{x} \\\\= x^{3/4}\cdot x^{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^{3/4}\cdot x^{1/2} \\\\= x^{\frac{3}{4}+\frac{1}{2}} \\\\= x^{\frac{3}{4}+\frac{2}{4}} \\\\= x^{\frac{5}{4}} .\end{array} 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 given expression is equivalent to \begin{array}{l}\require{cancel} x^{\frac{5}{4}} \\\\= \sqrt[4]{x^5} .\end{array} Extracting the factors that are perfect powers of the index, the expression above simplifies to \begin{array}{l}\require{cancel} \sqrt[4]{x^4\cdot x} \\\\= \sqrt[4]{(x)^4\cdot x} \\\\= x\sqrt[4]{x} .\end{array} Note that all variables are assumed to represent positive numbers.