Dear Student, (a) [{(625)-12}-14]2 ={(625)-12}-14×2 (as (am)n = amn) ={(625)-12}-12 =(625)-12×-12 (as (am)n = amn) =(625)14 =(252)14 =252×14 =2512 =5(b) (8116)-34[(259)-32÷(52)-3] =(3424)-34[(5232)-32÷(52)-3] =(32)4×-34[(53)2×-32÷(52)-3] (as (am)n = amn) =(32)-3[(53)-3 ÷(52)-3] =(23)3[(35)3 ÷(25)3] [as (ba)-n =( ab)n ] =(23)3[(35)3 ×(52)3] =827[27125×1258] =827×278 =1

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Dear Student,

(a) [{(625)^{\frac{- 1}{2}}}^{\frac{- 1}{4}}]^{2} = {(625)^{\frac{- 1}{2}}}^{\frac{- 1}{4} \times 2} (as (a^{m})^{n} = a^{mn}) = {(625)^{\frac{- 1}{2}}}^{\frac{- 1}{2}} = (625)^{\frac{- 1}{2} \times \frac{- 1}{2}} (as (a^{m})^{n} = a^{mn}) = (625)^{\frac{1}{4}} = (25^{2})^{\frac{1}{4}} = 25^{2 \times \frac{1}{4}} = 25^{\frac{1}{2}} = 5 (b) (\frac{81}{16})^{\frac{- 3}{4}} [(\frac{25}{9})^{\frac{- 3}{2}} \div (\frac{5}{2})^{- 3}] = (\frac{3^{4}}{2^{4}})^{\frac{- 3}{4}} [(\frac{5^{2}}{3^{2}})^{\frac{- 3}{2}} \div (\frac{5}{2})^{- 3}] = (\frac{3}{2})^{4 \times \frac{- 3}{4}} [(\frac{5}{3})^{\frac{2 \times - 3}{2}} \div (\frac{5}{2})^{- 3}] (as (a^{m})^{n} = a^{mn}) = (\frac{3}{2})^{- 3} [(\frac{5}{3})^{- 3} \div (\frac{5}{2})^{- 3}] = (\frac{2}{3})^{3} [(\frac{3}{5})^{3} \div (\frac{2}{5})^{3}] [as (\frac{b}{a})^{- n} = (\frac{a}{b})^{n}] = (\frac{2}{3})^{3} [(\frac{3}{5})^{3} \times (\frac{5}{2})^{3}] = \frac{8}{27} [\frac{27}{125} \times \frac{125}{8}] = \frac{8}{27} \times \frac{27}{8} = 1

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