The problem is to simplify expressions involving powers of quotients. We have 12 expressions to simplify.

1. Problem Description

The problem is to simplify expressions involving powers of quotients. We have 12 expressions to simplify.

2. Solution Steps

1) $(\frac{9h}{8h^5})^2$

$= \frac{(9h)^2}{(8h^5)^2}$

$= \frac{9^2 h^2}{8^2 (h^5)^2}$

$= \frac{81h^2}{64h^{10}}$

$= \frac{81}{64}h^{2-10}$

$= \frac{81}{64}h^{-8}$

$= \frac{81}{64h^8}$

2) $(\frac{5w^5 r^6}{3w^4 r^2})^2$

$= \frac{(5w^5 r^6)^2}{(3w^4 r^2)^2}$

$= \frac{5^2 (w^5)^2 (r^6)^2}{3^2 (w^4)^2 (r^2)^2}$

$= \frac{25w^{10}r^{12}}{9w^8 r^4}$

$= \frac{25}{9}w^{10-8}r^{12-4}$

$= \frac{25}{9}w^2 r^8$

3) $(\frac{5w^2}{4w^6 z^3})^2$

$= \frac{(5w^2)^2}{(4w^6 z^3)^2}$

$= \frac{5^2 (w^2)^2}{4^2 (w^6)^2 (z^3)^2}$

$= \frac{25w^4}{16w^{12}z^6}$

$= \frac{25}{16}w^{4-12}z^{-6}$

$= \frac{25}{16}w^{-8}z^{-6}$

$= \frac{25}{16w^8z^6}$

4) $(\frac{3n^4}{9n^6})^3$

$= \frac{(3n^4)^3}{(9n^6)^3}$

$= \frac{3^3 (n^4)^3}{9^3 (n^6)^3}$

$= \frac{27n^{12}}{729n^{18}}$

$= \frac{27}{729}n^{12-18}$

$= \frac{1}{27}n^{-6}$

$= \frac{1}{27n^6}$

5) $(\frac{7hg^3}{2h^6g^4})^2$

$= \frac{(7hg^3)^2}{(2h^6g^4)^2}$

$= \frac{7^2h^2(g^3)^2}{2^2(h^6)^2(g^4)^2}$

$= \frac{49h^2g^6}{4h^{12}g^8}$

$= \frac{49}{4}h^{2-12}g^{6-8}$

$= \frac{49}{4}h^{-10}g^{-2}$

$= \frac{49}{4h^{10}g^2}$

6) $(\frac{r^5}{r^6})^3$

$= (\frac{r^5}{r^6})^3 = (r^{5-6})^3 = (r^{-1})^3 = r^{-3} = \frac{1}{r^3}$

7) $(\frac{4^3}{4})^2$

$= (\frac{4^3}{4^1})^2$

$= (4^{3-1})^2$

$= (4^2)^2$

$= 4^4 = 256$

8) $(\frac{8w^3}{9w})^2$

$= \frac{(8w^3)^2}{(9w)^2}$

$= \frac{8^2 (w^3)^2}{9^2 w^2}$

$= \frac{64w^6}{81w^2}$

$= \frac{64}{81}w^{6-2}$

$= \frac{64}{81}w^4$

9) $(\frac{g}{g^2})^3$

$= (\frac{g^1}{g^2})^3 = (g^{1-2})^3 = (g^{-1})^3 = g^{-3} = \frac{1}{g^3}$

10) $(\frac{hk}{3h^6k^2})^3$

$= \frac{(hk)^3}{(3h^6k^2)^3}$

$= \frac{h^3k^3}{3^3(h^6)^3(k^2)^3}$

$= \frac{h^3k^3}{27h^{18}k^6}$

$= \frac{1}{27}h^{3-18}k^{3-6}$

$= \frac{1}{27}h^{-15}k^{-3}$

$= \frac{1}{27h^{15}k^3}$

11) $(\frac{6^2}{6^6})^2$

$= (6^{2-6})^2$

$= (6^{-4})^2$

$= 6^{-8}$

$= \frac{1}{6^8} = \frac{1}{1679616}$

12) $(\frac{8w^2s^5}{2ws^3})^2$

$= (\frac{8}{2}\frac{w^2}{w}\frac{s^5}{s^3})^2$

$= (4ws^2)^2$

$= 4^2 w^2 (s^2)^2$

$= 16w^2s^4$

3. Final Answer

1) $\frac{81}{64h^8}$

2) $\frac{25}{9}w^2r^8$

3) $\frac{25}{16w^8z^6}$

4) $\frac{1}{27n^6}$

5) $\frac{49}{4h^{10}g^2}$

6) $\frac{1}{r^3}$

7) $256$

8) $\frac{64}{81}w^4$

9) $\frac{1}{g^3}$

10) $\frac{1}{27h^{15}k^3}$

11) $\frac{1}{1679616}$

12) $16w^2s^4$