Calculate: 3x^{-2}=27

Expression: $3{x}^{-2}=27$

Determine the defined range

$\begin{array} { l }3{x}^{-2}=27,& x≠0\end{array}$

Express with a positive exponent using ${a}^{-n}=\frac{ 1 }{ {a}^{n} }$

$3 \times \frac{ 1 }{ {x}^{2} }=27$

Calculate the product

$\frac{ 3 }{ {x}^{2} }=27$

Multiply both sides of the equation by ${x}^{2}$

$3=27{x}^{2}$

Swap the sides of the equation

$27{x}^{2}=3$

Divide both sides of the equation by $27$

${x}^{2}=\frac{ 1 }{ 9 }$

Take the square root of both sides of the equation and remember to use both positive and negative roots

$x=\frac{ 1 }{ 3 }$

Write the solutions, one with a $+$ sign and one with a $-$ sign

$\begin{array} { l }\begin{array} { l }x=-\frac{ 1 }{ 3 },\\x=\frac{ 1 }{ 3 }\end{array},& x≠0\end{array}$

Check if the solution is in the defined range

$\begin{array} { l }x=-\frac{ 1 }{ 3 },\\x=\frac{ 1 }{ 3 }\end{array}$

The equation has $2$ solutions

$\begin{align*}&\begin{array} { l }x_1=-\frac{ 1 }{ 3 },& x_2=\frac{ 1 }{ 3 }\end{array} \\&\begin{array} { l }x_1\approx-0.333333,& x_2\approx0.333333\end{array}\end{align*}$