Express each of the following rational numbers with a positive exponen

1.	Express each of the following rational numbers with a positive exponent:(i) \((\frac{3}{4})^{-2}\)(ii) \((\frac{5}{4})^{-3}\)(iii) \(4^{3}\times 4^{-9}\)(iv) \(\{(\frac{4}{3})^{-3}\}^{-4}\)(v) \(\{(\frac{3}{2})^{4}\}^{-2}\)
Answer» (i) \((\frac{3}{4})^{-2}\) ⇒ \((\frac{3}{4})^{-2}\)= \((\frac{4}{3})^{2}\)[Using \((\frac{a}{b})^{-n}\)= \((\frac{b}{a})^{n}\)] (ii) \((\frac{5}{4})^{-3}\) ⇒ \((\frac{5}{4})^{-3}\)= \((\frac{4}{5})^{a}\)[Using \((\frac{a}{b})^{-n}\)= \((\frac{b}{a})^{n}\) ] (iii) \(4^{3}\times 4^{-9}\) ⇒ \(4^{3}\times 4^{-9}\)= \(4^{3-9}\)= \(4^{-6}\)[Using (aⁿ× aᵐ = aᵐ⁺ⁿ ] ⇒ \(4^{-6}\)= \((\frac{1}{4})^{6}\)[Using \(\frac{1}{a^{n}}\)= \(a^{-n}\)] (iv) \(\{(\frac{4}{3})^{-3}\}^{-4}\) ⇒ \(\{(\frac{4}{3})^{-3}\}^{-4}\)= \((\frac{4}{3})^{12}\)[Using (aⁿ)ᵐ = aᵐⁿ ] (v) \(\{(\frac{3}{2})^{4}\}^{-2}\) ⇒ \(\{(\frac{3}{2})^{4}\}^{-2}\)= \((\frac{3}{2})^{-8}\)= \((\frac{2}{3})^{8}\)[Using (aⁿ)ᵐ = aᵐⁿ and \(\frac{1}{a^{n}}\)= \(a^{-n}\)]

Express each of the following rational numbers with a positive exponent:(i) \((\frac{3}{4})^{-2}\)(ii) \((\frac{5}{4})^{-3}\)(iii) \(4^{3}\times 4^{-9}\)(iv) \(\{(\frac{4}{3})^{-3}\}^{-4}\)(v) \(\{(\frac{3}{2})^{4}\}^{-2}\)

Answer»

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

⇒ \((\frac{3}{4})^{-2}\)= \((\frac{4}{3})^{2}\)[Using \((\frac{a}{b})^{-n}\)= \((\frac{b}{a})^{n}\)]

(ii) \((\frac{5}{4})^{-3}\)

⇒ \((\frac{5}{4})^{-3}\)= \((\frac{4}{5})^{a}\)[Using \((\frac{a}{b})^{-n}\)= \((\frac{b}{a})^{n}\) ]

(iii) \(4^{3}\times 4^{-9}\)

⇒ \(4^{3}\times 4^{-9}\)= \(4^{3-9}\)= \(4^{-6}\)[Using (aⁿ× aᵐ = aᵐ⁺ⁿ ]

⇒ \(4^{-6}\)= \((\frac{1}{4})^{6}\)[Using \(\frac{1}{a^{n}}\)= \(a^{-n}\)]

(iv) \(\{(\frac{4}{3})^{-3}\}^{-4}\)

⇒ \(\{(\frac{4}{3})^{-3}\}^{-4}\)= \((\frac{4}{3})^{12}\)[Using (aⁿ)ᵐ = aᵐⁿ ]

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

⇒ \(\{(\frac{3}{2})^{4}\}^{-2}\)= \((\frac{3}{2})^{-8}\)= \((\frac{2}{3})^{8}\)[Using (aⁿ)ᵐ = aᵐⁿ and \(\frac{1}{a^{n}}\)= \(a^{-n}\)]