Express each of the following rational numbers with a positive exponent: (i) (3/4)^-2 (ii) (5/4)^-3 (iii) 4^3x4^-9

Question

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 1

(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 \(\frac{1}{a^{n}}\)= \(a^{-n}\)]