Answer
The solutions are $x=2$ and $x=1$
Work Step by Step
$(x^{2}-3x+3)^{3/2}-1=0$
Take $1$ to the right side of the equation:
$(x^{2}-3x+3)^{3/2}=1$
Raise both sides to $\dfrac{2}{3}$:
$[(x^{2}-3x+3)^{3/2}]^{2/3}=1^{2/3}$
$x^{2}-3x+3=1$
Take $1$ to the left side and simplify:
$x^{2}-3x+3-1=0$
$x^{2}-3x+2=0$
Solve by factoring:
$(x-2)(x-1)=0$
Set both factors equal to $0$ and solve each individual equation for $x$:
$x-2=0$
$x=2$
$x-1=0$
$x=1$
The solutions are $x=2$ and $x=1$
