Express each of the following rational numbers with a negative exponen

1.	Express each of the following rational numbers with a negative exponent:(i) \((\frac{1}{4})^{3}\)(ii) \(3^{5}\)(iii) \((\frac{3}{5})^{4}\)(iv) \(\{(\frac{3}{2})^{4}\}^{-3}\)(v) \(\{(\frac{7}{3})^{4}\}^{-3}\)
Answer» (i) \((\frac{1}{4})^{3}\) ⇒ \((\frac{1}{4})^{3}\)= \((4)^{-3}\)[Using \(\frac{1}{a^{n}}\)= \(a^{-n}\)] (ii) \(3^{5}\) ⇒ \(3^{5}\)= \((\frac{1}{3})^{-5}\)= \((4)^{-3}\)[Using \(\frac{1}{a^{n}}\)= \(a^{-n}\)] (iii) \((\frac{3}{5})^{4}\) ⇒ \((\frac{3}{5})^{4}\)= \((\frac{5}{3})^{-4}\)[Using \((\frac{a}{b})^{-n}\)= \((\frac{b}{n})^{n}\)] (iv) \(\{(\frac{3}{2})^{4}\}^{-3}\) ⇒ \(\{(\frac{3}{2})^{4}\}^{-3}\) = \((\frac{3}{2})^{-12}\)[Using (aⁿ)ᵐ = aᵐⁿ] (v) \(\{(\frac{7}{3})^{4}\}^{-3}\) ⇒ \(\{(\frac{7}{3})^{4}\}^{-3}\)= \((\frac{7}{3})^{-12}\)[Using (aⁿ)ᵐ = aᵐⁿ]

Express each of the following rational numbers with a negative exponent:(i) \((\frac{1}{4})^{3}\)(ii) \(3^{5}\)(iii) \((\frac{3}{5})^{4}\)(iv) \(\{(\frac{3}{2})^{4}\}^{-3}\)(v) \(\{(\frac{7}{3})^{4}\}^{-3}\)

Answer»

(i) \((\frac{1}{4})^{3}\)

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

(ii) \(3^{5}\)

⇒ \(3^{5}\)= \((\frac{1}{3})^{-5}\)= \((4)^{-3}\)[Using \(\frac{1}{a^{n}}\)= \(a^{-n}\)]

(iii) \((\frac{3}{5})^{4}\)

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

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

⇒ \(\{(\frac{3}{2})^{4}\}^{-3}\) = \((\frac{3}{2})^{-12}\)[Using (aⁿ)ᵐ = aᵐⁿ]

(v) \(\{(\frac{7}{3})^{4}\}^{-3}\)

⇒ \(\{(\frac{7}{3})^{4}\}^{-3}\)= \((\frac{7}{3})^{-12}\)[Using (aⁿ)ᵐ = aᵐⁿ]

Simplify:(i) (4-1 × 3-1)2(ii) (5-1 ÷ 6-1)3(iii) (2-1 + 3-1)-1
Write each of the following in exponential form:(i) \((\frac{3}{2})^{-1}\times (\frac{3}{2})^{-1}\times (\frac{3}{2})^{-1}\times (\frac{3}{2})^{-1}\)(ii) \((\frac{2}{5})^{-2}\times (\frac{2}{5})^{-2}\times (\frac{2}{5})^{-2}\)
By what number should (-15)-1 be divided so that the quotient may be equal to (-5)-1 ?
By what number should \((-15)^{-1}\) be divided so that the quotient may be equal to \((-5)^{-1}\)?
By what number should (1/2)-1 be multiplied so that the product may be equal to (-4/7)-1 ?
Find x, if(i) \((\frac{1}{4})^{-4}\times (\frac{1}{4})^{-8}=(\frac{1}{4})^{-4x}\)(ii) \((\frac{-1}{2})^{-19}\div (\frac{-1}{2})^{8}=(\frac{-1}{2})^{-2x+1}\)(iii) \((\frac{3}{2})^{-3}\times (\frac{3}{2})^{5}=(\frac{3}{2})^{-2x+1}\)(iv) \((\frac{2}{5})^{-3}\times (\frac{2}{5})^{15}=(\frac{2}{5})^{2+3x}\)(v) \((\frac{5}{4})^{-x}\div (\frac{5}{4})^{-4}=(\frac{5}{4})^{5}\)(vi) \((\frac{8}{3})^{2x+1}\times (\frac{8}{3})^{5}=(\frac{8}{3})^{x+2}\)
By what number should \(5^{-1}\) be multiplied so that the product may be equal to \((-7)^{-1}\) ?
By what number should 5-1 be multiplied so that the product may be equal to (-7)-1 ?
Find x, if(i) (1/4)-4 × (1/4)-8 = (1/4)-4x(ii) (-1/2)-19 ÷ (-1/2)8 = (-1/2)-2x+1(iii) (3/2)-3 × (3/2)5 = (3/2)2x+1
By what number should \((\frac{5}{3})^{-2}\) be multiplied so that the product may be \((\frac{7}{3})^{-1}\)?