Answer
$\sqrt[12]{y^{7}}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To simplify the given expression, $
\sqrt[3]{y}\cdot\sqrt[4]{y}
,$ express the radicals as radicals with same indices by finding the $LCD$ of the indices. Once the indices are the same, use the laws of radicals to simplify the expression.
$\bf{\text{Solution Details:}}$
The $LCD$ of the indices, $
3
$ and $
4
,$ is $
12
$ since it is the lowest number that can be divided exactly by both indices. Multiplying the index by a number to make it equal to the $LCD$ and raising the radicand by the same multiplier results to
\begin{array}{l}\require{cancel}
\sqrt[3(4)]{y^4}\cdot\sqrt[4(3)]{y^3}
\\\\=
\sqrt[12]{y^4}\cdot\sqrt[12]{y^3}
.\end{array}
Using the Product Rule of radicals which is given by $\sqrt[m]{x}\cdot\sqrt[m]{y}=\sqrt[m]{xy},$ the expression above is equivalent to\begin{array}{l}\require{cancel}
\sqrt[12]{y^4(y^3)}
.\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}
\sqrt[12]{y^{4+3}}
\\\\=
\sqrt[12]{y^{7}}
.\end{array}