(i) \(\{(\frac{1}{3})^{-3}-(\frac{1}{2})^{-3}\}\div (\frac{1}{4})^{-3}\)

⇒ \(\{(\frac{1}{3})^{-3}-(\frac{1}{2})^{-3}\}\div (\frac{1}{4})^{-3}\)

⇒ \((3^{3}-2^{3})\div 4^{3}\)[Using \(\cfrac{1}{a^{n}}\)= \(a^{-n}\)] 

⇒ (27-8) ÷ \(4^{3}\) 

⇒ 19 ÷ \(4^{3}\)

⇒ \(19\times \frac{1}{64}\)= \(\frac{19}{64}\)[Using and \(\frac{1}{a}\div \frac{1}{b}\)= \(\frac{1}{a}\times \frac{b}{1}\)] 

(ii) \((3^{2}-2^{2})\times (\frac{2}{3})^{-3}\)

⇒ \((3^{2}-2^{2})\times (\frac{2}{3})^{-3}\)

⇒ \((9-4)\times (\frac{3}{2})^{-3}\)[Using  \(\cfrac{1}{a^{n}}\)= \(a^{-n}\)] 

⇒ \((5)\times (\frac{3}{2})^{3}\)

⇒ \((5)\times (\frac{27}{8})\)[Using \(a^{n}=a\times a ...... ........n\,times\)] 

⇒ \(\frac{135}{8}\)

(iii) \(\{(\frac{1}{2})^{-1}\times (-4)^{-1}\}^{-1}\)

⇒ \(\{(\frac{1}{2})^{-1}\times (-4)^{-1}\}^{-1}\)

⇒ \(\{(2)\times (\frac{1}{-4})\}^{-1}\)[Using  \(\cfrac{1}{a^{n}}\)= \(a^{-n}\)] 

⇒ \(\{-\frac{1}{2}\}^{-1}\)

⇒ -2

(iv) \([\{(\frac{-1}{4})^{2}\}^{-2}]^{-1}\)

⇒ \([\{(\frac{-1}{4})^{2}\}^{-2}]^{-1}\)

⇒ \(\{-\frac{1}{4}\}^{4}\)[Using (aⁿ)ᵐ = aᵐⁿ] 

⇒ \(\frac{1}{256}\)[Using \(a^{n}=a\times a ...... ........n\,times\)] 

(v) \(\{(\frac{2}{3})^{2}\}^{3}\times (\frac{1}{3})^{-4}\times 3^{-1}\times 6^{-1}\)

\((\frac{2}{3})^{6}\times (\frac{1}{3})^{-4}\times \frac{1}{3}\times \frac{1}{2\times 3}\)

\((\frac{2}{3})^{6}\times (\frac{1}{3})^{-4}\times \frac{1}{3}\times \frac{1}{2}\times \frac{1}{3}\)

\(\cfrac{2^{6-1}}{3^{6-4+1+1}}\)      \(a^{m}\times a^{n}=a^{m+n}\)

\(\cfrac{2^{5}}{3^{4}}\)= \(\cfrac{32}{81}\)