#### Answer

$\left\{ -\dfrac{4}{3},0,\dfrac{4}{3} \right\}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
Express the given equation, $
9x^3=16x
,$ in factored form. Then use the Zero Product Property by equating each factor to zero. Finally, solve each of the resulting equations.
$\bf{\text{Solution Details:}}$
Using the properties of equality, the given equation is equivalent to
\begin{array}{l}\require{cancel}
9x^3-16x=0
.\end{array}
The $GCF$ of the constants of the terms $\{
9,-16
\}$ is $
1
$ since it is the highest number that can divide all the given constants. The $GCF$ of the common variable/s is the variable/s with the lowest exponent. Hence, the $GCF$ of the common variable/s $\{
x^3,x
\}$ is $
x
.$ Hence, the entire expression has $GCF=
x
.$
Factoring the $GCF=
x
,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
x(9x^2-16)=0
.\end{array}
The expressions $
9x^2
$ and $
16
$ are both perfect squares and are separated by a minus sign. Hence, $
9x^2-16
,$ is a difference of $2$ squares. Using the factoring of the difference of $2$ squares which is given by $a^2-b^2=(a+b)(a-b),$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
x(3x+4)(3x-4)=0
.\end{array}
Equating each factor to zero (Zero Product Property), then
\begin{array}{l}\require{cancel}
x=0
\\\\\text{OR}\\\\
3x+4=0
\\\\\text{OR}\\\\
3x-4=0
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
x=0
\\\\\text{OR}\\\\
3x+4=0
\\\\
3x=-4
\\\\
x=-\dfrac{4}{3}
\\\\\text{OR}\\\\
3x-4=0
\\\\
3x=4
\\\\
x=\dfrac{4}{3}
.\end{array}
Hence, the solutions are $
\left\{ -\dfrac{4}{3},0,\dfrac{4}{3} \right\}
.$