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

AlgebraExponentsSimplificationAlgebraic ExpressionsPowers
2025/4/24

1. Problem Description

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

2. Solution Steps

1) (9h8h5)2(\frac{9h}{8h^5})^2
=(9h)2(8h5)2= \frac{(9h)^2}{(8h^5)^2}
=92h282(h5)2= \frac{9^2 h^2}{8^2 (h^5)^2}
=81h264h10= \frac{81h^2}{64h^{10}}
=8164h2−10= \frac{81}{64}h^{2-10}
=8164h−8= \frac{81}{64}h^{-8}
=8164h8= \frac{81}{64h^8}
2) (5w5r63w4r2)2(\frac{5w^5 r^6}{3w^4 r^2})^2
=(5w5r6)2(3w4r2)2= \frac{(5w^5 r^6)^2}{(3w^4 r^2)^2}
=52(w5)2(r6)232(w4)2(r2)2= \frac{5^2 (w^5)^2 (r^6)^2}{3^2 (w^4)^2 (r^2)^2}
=25w10r129w8r4= \frac{25w^{10}r^{12}}{9w^8 r^4}
=259w10−8r12−4= \frac{25}{9}w^{10-8}r^{12-4}
=259w2r8= \frac{25}{9}w^2 r^8
3) (5w24w6z3)2(\frac{5w^2}{4w^6 z^3})^2
=(5w2)2(4w6z3)2= \frac{(5w^2)^2}{(4w^6 z^3)^2}
=52(w2)242(w6)2(z3)2= \frac{5^2 (w^2)^2}{4^2 (w^6)^2 (z^3)^2}
=25w416w12z6= \frac{25w^4}{16w^{12}z^6}
=2516w4−12z−6= \frac{25}{16}w^{4-12}z^{-6}
=2516w−8z−6= \frac{25}{16}w^{-8}z^{-6}
=2516w8z6= \frac{25}{16w^8z^6}
4) (3n49n6)3(\frac{3n^4}{9n^6})^3
=(3n4)3(9n6)3= \frac{(3n^4)^3}{(9n^6)^3}
=33(n4)393(n6)3= \frac{3^3 (n^4)^3}{9^3 (n^6)^3}
=27n12729n18= \frac{27n^{12}}{729n^{18}}
=27729n12−18= \frac{27}{729}n^{12-18}
=127n−6= \frac{1}{27}n^{-6}
=127n6= \frac{1}{27n^6}
5) (7hg32h6g4)2(\frac{7hg^3}{2h^6g^4})^2
=(7hg3)2(2h6g4)2= \frac{(7hg^3)^2}{(2h^6g^4)^2}
=72h2(g3)222(h6)2(g4)2= \frac{7^2h^2(g^3)^2}{2^2(h^6)^2(g^4)^2}
=49h2g64h12g8= \frac{49h^2g^6}{4h^{12}g^8}
=494h2−12g6−8= \frac{49}{4}h^{2-12}g^{6-8}
=494h−10g−2= \frac{49}{4}h^{-10}g^{-2}
=494h10g2= \frac{49}{4h^{10}g^2}
6) (r5r6)3(\frac{r^5}{r^6})^3
=(r5r6)3=(r5−6)3=(r−1)3=r−3=1r3= (\frac{r^5}{r^6})^3 = (r^{5-6})^3 = (r^{-1})^3 = r^{-3} = \frac{1}{r^3}
7) (434)2(\frac{4^3}{4})^2
=(4341)2= (\frac{4^3}{4^1})^2
=(43−1)2= (4^{3-1})^2
=(42)2= (4^2)^2
=44=256= 4^4 = 256
8) (8w39w)2(\frac{8w^3}{9w})^2
=(8w3)2(9w)2= \frac{(8w^3)^2}{(9w)^2}
=82(w3)292w2= \frac{8^2 (w^3)^2}{9^2 w^2}
=64w681w2= \frac{64w^6}{81w^2}
=6481w6−2= \frac{64}{81}w^{6-2}
=6481w4= \frac{64}{81}w^4
9) (gg2)3(\frac{g}{g^2})^3
=(g1g2)3=(g1−2)3=(g−1)3=g−3=1g3= (\frac{g^1}{g^2})^3 = (g^{1-2})^3 = (g^{-1})^3 = g^{-3} = \frac{1}{g^3}
10) (hk3h6k2)3(\frac{hk}{3h^6k^2})^3
=(hk)3(3h6k2)3= \frac{(hk)^3}{(3h^6k^2)^3}
=h3k333(h6)3(k2)3= \frac{h^3k^3}{3^3(h^6)^3(k^2)^3}
=h3k327h18k6= \frac{h^3k^3}{27h^{18}k^6}
=127h3−18k3−6= \frac{1}{27}h^{3-18}k^{3-6}
=127h−15k−3= \frac{1}{27}h^{-15}k^{-3}
=127h15k3= \frac{1}{27h^{15}k^3}
11) (6266)2(\frac{6^2}{6^6})^2
=(62−6)2= (6^{2-6})^2
=(6−4)2= (6^{-4})^2
=6−8= 6^{-8}
=168=11679616= \frac{1}{6^8} = \frac{1}{1679616}
12) (8w2s52ws3)2(\frac{8w^2s^5}{2ws^3})^2
=(82w2ws5s3)2= (\frac{8}{2}\frac{w^2}{w}\frac{s^5}{s^3})^2
=(4ws2)2= (4ws^2)^2
=42w2(s2)2= 4^2 w^2 (s^2)^2
=16w2s4= 16w^2s^4

3. Final Answer

1) 8164h8\frac{81}{64h^8}
2) 259w2r8\frac{25}{9}w^2r^8
3) 2516w8z6\frac{25}{16w^8z^6}
4) 127n6\frac{1}{27n^6}
5) 494h10g2\frac{49}{4h^{10}g^2}
6) 1r3\frac{1}{r^3}
7) 256256
8) 6481w4\frac{64}{81}w^4
9) 1g3\frac{1}{g^3}
10) 127h15k3\frac{1}{27h^{15}k^3}
11) 11679616\frac{1}{1679616}
12) 16w2s416w^2s^4

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