#### Answer

$m=\left\{ 6-\sqrt{3},6+\sqrt{3} \right\}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given equation, $
(m-6)^2=27
,$ take the square root of both sides (Square Root Property) and simplify the radical. Then use the properties of equality to isolate the variable.
$\bf{\text{Solution Details:}}$
Taking the square root of both sides (Square Root Property), the equation above is equivalent to
\begin{array}{l}\require{cancel}
m-6=\pm\sqrt{27}
.\end{array}
Simplifying the radical and using the properties of equality to isolate the variable, the equation above is equivalent to
\begin{array}{l}\require{cancel}
m-6=\pm\sqrt{9\cdot3}
\\\\
m-6=\pm\sqrt{(3)^2\cdot3}
\\\\
m-6=\pm3\sqrt{3}
\\\\
m=6\pm3\sqrt{3}
.\end{array}
The solutions are
\begin{array}{l}\require{cancel}
m=6-\sqrt{3}
\\\\\text{OR}\\\\
m=6+\sqrt{3}
.\end{array}
Hence, $
m=\left\{ 6-\sqrt{3},6+\sqrt{3} \right\}
.$