[latex]1.\left(frac{4^{2n+1}cdot8^{n-1}cdot2^n}{16^{2n+1}} ight)^n=left(frac{(2^2)^{2n+1}cdot(2^3)^{n-1}cdot2^n}{(2^4)^{2n+1}} ight)^n=left(frac{2^{2(2n+1)}cdot2^{3(n-1)}cdot2^n}{2^{4(2n+1)}} ight)^n\\=left(frac{2^{4n+2}cdot2^{3n-3}cdot2^n}{2^{8n+4}} ight)^n=left(frac{2^{4n+2+3n-3+n}}{2^{8n+4}} ight)^n=left(frac{2^{8n-1}}{2^{8n+4}} ight)^n=left(2^{8n-1-(8n+4)} ight)^n\\=left(2^{8n-1-8n-4} ight)^n=left(2^{-5} ight)^n=2^{-5n}[/latex] ============================================================== [latex]2.\frac{left(2ab ight)^frac{2}{3}cdotleft(2a ight)^{-3}cdotleft(3b ight)^{-frac{5}{3}}}{left(3^frac{1}{2}cdot a ight)^frac{4}{3}cdotleft(2ab ight)^{-2}}=frac{2^frac{2}{3}a^frac{2}{3}b^frac{2}{3}cdot2^{-3}a^{-3}cdot3^{-frac{5}{3}}b^{-frac{5}{3}}}{left(3^frac{1}{2} ight)^frac{4}{3}a^frac{4}{3}cdot2^{-2}a^{-2}b^{-2}}[/latex] [latex]=2^{frac{2}{3}-3-(-2)}cdot3^{-frac{5}{3}-(-frac{2}{3})}cdot a^{frac{2}{3}-3-(-frac{4}{3})-(-2)}cdot b^{frac{2}{3}-frac{5}{3}-(-2)}\\=2^{-frac{1}{3}}cdot3^{-1}cdot a^{-1}cdot b^1=left(frac{1}{2} ight)^frac{1}{3}cdotfrac{1}{3}cdotfrac{1}{a}cdot b=frac{b}{3sqrt[3]2 a}[/latex] [latex]frac{b}{3sqrt[3]2 a}cdotfrac{sqrt[3]4}{sqrt[3]4}=frac{bsqrt[3]4}{3sqrt[3]8 a}=frac{bsqrt[3]4}{3cdot2a}=frac{bsqrt[3]4}{6a}[/latex] ============================================================ [latex]3.\left(frac{left[left(-4frac{1}{5} ight)^2 ight]^{-4}cdotleft[left(-4,2 ight)^{-3} ight]^2}{left(-frac{21}{5} ight)^{-12}} ight)^{-1}=frac{left(4frac{1}{5} ight)^{-12}}{left(4frac{1}{5} ight)^{2cdot(-4)}cdotleft(4frac{1}{5} ight)^{-3cdot2}}=frac{left(4frac{1}{5} ight)^{-12}}{left(4frac{1}{5} ight)^{-8}cdotleft(4frac{1}{5} ight)^{-6}}[/latex] [latex]=frac{left(4frac{1}{5} ight)^{-12}}{left(4frac{1}{5} ight)^{-8-6}}=frac{left(4frac{1}{5} ight)^{-12}}{left(4frac{1}{5} ight)^{-14}}=left(4frac{1}{5} ight)^{-12-(-14)}=left(4frac{1}{5} ight)^2=left(frac{21}{5} ight)^2=frac{441}{25}=17,64[/latex]
