Answer
$x^{17/20}$
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[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}
