Answer
$\left\{1,\sqrt[3]{9}\right\}$
Work Step by Step
Using the properties of equality, the given equation, $
x^6-10x^3=-9
,$ is equivalent to
\begin{align*}
x^6-10x^3+9&=0
.\end{align*}
Using factoring of trinomials, the equation above is equivalent to
\begin{align*}
(x^3-9)(x^3-1)&=0
.\end{align*}
Equating each factor to zero (Zero Product Property) and solving for the variable, then
\begin{array}{l|r}
x^3-9=0 & x^3-1=0
\\
x^3=9 & x^3=1
.\end{array}
Taking the cube root of both sides, the equations above are equivalent to
\begin{array}{l|r}
x=\sqrt[3]{9} & x=\sqrt[3]{1}
\\
& x=1
.\end{array}
Hence, the solution set of the equation $
x^6-10x^3=-9
$ is $\left\{1,\sqrt[3]{9}\right\}$.
