To solve the expression [tex]\(\left( \frac{2^3 p^{-4} g^5 r^2}{2 g^{-3} p r^4} \right)^{-1}\)[/tex], follow the steps below:
1. Simplify the Inner Expression:
[tex]\[
\frac{2^3 p^{-4} g^5 r^2}{2 g^{-3} p r^4}
\][/tex]
2. Simplify the Constants:
- The constants in the expression are [tex]\(2^3\)[/tex] and [tex]\(2\)[/tex]:
[tex]\[
\frac{2^3}{2} = \frac{8}{2} = 4
\][/tex]
3. Simplify the Variables Separately:
- For [tex]\(p\)[/tex]:
[tex]\[
\frac{p^{-4}}{p} = p^{-4-1} = p^{-5}
\][/tex]
- For [tex]\(g\)[/tex]:
[tex]\[
\frac{g^5}{g^{-3}} = g^{5+3} = g^8
\][/tex]
- For [tex]\(r\)[/tex]:
[tex]\[
\frac{r^2}{r^4} = r^{2-4} = r^{-2}
\][/tex]
4. Combine the Simplified Parts:
- Combine the constants and the simplified variables:
[tex]\[
4 \cdot p^{-5} \cdot g^8 \cdot r^{-2} = \frac{4 g^8}{p^5 r^2}
\][/tex]
5. Inverse the Expression:
- Now, take the inverse of the simplified expression:
[tex]\[
\left( \frac{4 g^8}{p^5 r^2} \right)^{-1} = \frac{p^5 r^2}{4 g^8}
\][/tex]
6. Final Simplified Expression:
- The final simplified expression after taking the inverse is:
[tex]\[
\frac{p^5 r^2}{4 g^8}
\][/tex]
Thus, the fully simplified result of the given expression is:
[tex]\[
\left( \frac{2^3 p^{-4} g^5 r^2}{2 g^{-3} p r^4} \right)^{-1} = \frac{p^5 r^2}{4 g^8}
\][/tex]
