Simplify the expression. Show the work

${\left(\frac{{\displaystyle -2r^{-2}{q}^5{t}^2}}{{\displaystyle 5r^4{q}^2{t}^{-3}}}\right)}^{\mathbf{-}\mathbf{2}}$

If a fraction is raised to a power, then every terms of the fraction is raised to the power.

In general:${\left(\frac{{\displaystyle x}}{{\displaystyle y}}\right)}^m=\frac{x^m}{y^m}$

If the two terms having the same base are divided, then the base is written only once and the indices are subtracted.

In general: $\frac{x^m}{x^n}=x^{m-n}$

If a term with a power is itself raised to a power, then the powers are multiplied together.

In general: ${\left(x^m\right)}^n=x^{m\times n}$

If a term has negative index, then the reciprocal of that term is taken and raise it with the positive index.

In general:$x^{-n}=\frac{1}{x^n}$

Using laws of indices simplify${\left(\frac{{\displaystyle -2r^{-2}{q}^5{t}^2}}{{\displaystyle 5r^4{q}^2{t}^{-3}}}\right)}^{-2}$

$\begin{array}{l}{\left(\frac{-2r^{-2}{q}^5{t}^2}{5r^4{q}^2{t}^{-3}}\right)}^{-2}={\left[\left(\frac{-2}{5}\right)\left(\frac{r{}^{-2}}{r^4}\right)\left(\frac{q^5}{q^2}\right)\left(\frac{t^2}{t^{-3}}\right)\right]}^{-2}\left[\text{Collect\hspace{0.17em}\hspace{0.17em}the\hspace{0.17em}\hspace{0.17em}like\hspace{0.17em}\hspace{0.17em}terms\hspace{0.17em}\hspace{0.17em}together}\right]\\ ={\left[\left(\frac{-2}{5}\right)\left(r^{-2-4}\right)\left(q^{5-2}\right)\left(t^{2-(-3)}\right)\right]}^{-2}[\text{Using}\frac{x^m}{x^n}=x^{m-n}]\\ ={\left[\left(\frac{-2}{5}\right)\left(r^{-6}\right)\left(q^3\right)\left(t^{2+3}\right)\right]}^{-2}\\ ={\left[\left(\frac{-2}{5}\right)\left(r^{-6}\right)\left(q^3\right)\left(t^5\right)\right]}^{-2}\\ ={\left(\frac{-2}{5}\right)}^{-2}{\left(r^{-6}\right)}^{-2}{\left(q^3\right)}^{-2}{\left(t^5\right)}^{-2}[\text{Using\hspace{0.17em}\hspace{0.17em}}{\left(xy\right)}^m=x^m{y}^m]\\ ={\left(\frac{-2}{5}\right)}^{-2}\left(r^{(-6)(-2)}\right)\left(q^{\left(3\right)(-2)}\right)\left(t^{\left(5\right)(-2)}\right)[\text{Using\hspace{0.17em}\hspace{0.17em}}{\left(x^m\right)}^n=x^{m\times n}]\\ ={\left(\frac{-2}{5}\right)}^{-2}\left(r^{12}\right)\left(q^{-6}\right)\left(t^{-10}\right)\\ ={\left(\frac{5}{-2}\right)}^2\left(r^{12}\right)\left(\frac{1}{q^6}\right)\left(\frac{1}{t^{10}}\right)[\text{Using\hspace{0.17em}\hspace{0.17em}}{x}^{-n}=\frac{1}{x^n}]\\ =\frac{{\left(5\right)}^2}{{(-2)}^2}\times \frac{r^{12}}{q^6{t}^{10}}[\text{Using\hspace{0.17em}\hspace{0.17em}}{\left(\frac{x}{y}\right)}^m=\frac{x^m}{y^m}]\\ =\frac{25}{4}\times \frac{r^{12}}{q^6{t}^{10}}\\ =\frac{25r^{12}}{4q^6{t}^{10}}\end{array}$