Intermediate Algebra (12th Edition)

$x^{17/20}$
$\bf{\text{Solution Outline:}}$ Use the definition of rational exponents and the laws of exponents to simplify the given expression, $\sqrt[5]{x^3}\cdot\sqrt[4]{x} .$ $\bf{\text{Solution Details:}}$ 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} x^{3/5}\cdot x^{1/4} .\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{3}{5}+\frac{1}{4}} .\end{array} To simplify the expression $\dfrac{3}{5}+\dfrac{1}{4} ,$ change the expressions to similar fractions (same denominator) by using the $LCD$. The $LCD$ of the denominators $5$ and $4$ is $20$ 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{3}{5}\cdot\frac{4}{4}+\frac{1}{4}\cdot\frac{5}{5}} \\\\= x^{\frac{12}{20}+\frac{5}{20}} \\\\= x^{\frac{17}{20}} \\\\= x^{17/20} .\end{array}