Sure! Let's simplify the given expression step-by-step.
The expression to simplify is [tex]\(\left[\left\{2 p^{-1} q^2 r\right\}^3\right]^{-2}\)[/tex].
1. Simplify inside the curly braces [tex]\(\left\{2 p^{-1} q^2 r\right\}\)[/tex]:
[tex]\[
2 p^{-1} q^2 r
\][/tex]
2. Raise the entire expression inside the curly braces to the power of 3:
[tex]\[
\left(2 p^{-1} q^2 r\right)^3
\][/tex]
Apply the exponent to each factor inside the parentheses:
[tex]\[
2^3 \cdot (p^{-1})^3 \cdot (q^2)^3 \cdot r^3
\][/tex]
3. Calculate the exponents for each factor:
[tex]\[
2^3 = 8, \quad (p^{-1})^3 = p^{-3}, \quad (q^2)^3 = q^6, \quad r^3 = r^3
\][/tex]
4. Combine these results:
[tex]\[
8 \cdot p^{-3} \cdot q^6 \cdot r^3
\][/tex]
This simplifies to:
[tex]\[
8 p^{-3} q^6 r^3
\][/tex]
5. Raise the entire resulting expression to the power of -2:
[tex]\[
\left(8 p^{-3} q^6 r^3\right)^{-2}
\][/tex]
Apply the exponent to each factor inside the parentheses:
[tex]\[
8^{-2} \cdot (p^{-3})^{-2} \cdot (q^6)^{-2} \cdot (r^3)^{-2}
\][/tex]
6. Calculate the exponents for each factor:
[tex]\[
8^{-2} = \frac{1}{8^2} = \frac{1}{64}
\][/tex]
[tex]\[
(p^{-3})^{-2} = p^{6}
\][/tex]
[tex]\[
(q^6)^{-2} = q^{-12}
\][/tex]
[tex]\[
(r^3)^{-2} = r^{-6}
\][/tex]
7. Combine these results:
[tex]\[
\frac{1}{64} \cdot p^6 \cdot q^{-12} \cdot r^{-6}
\][/tex]
8. Simplify the expression by combining the powers:
[tex]\[
\frac{p^6}{64 q^{12} r^6}
\][/tex]
Thus, the simplified expression is:
[tex]\[
\boxed{\frac{p^6}{64 q^{12} r^6}}
\][/tex]
