Question

Express each of the following as a rational number:

(i) ${\left(\frac{2}{3}\right)}^5$
(ii) ${\left(\frac{-8}{5}\right)}^3$
(iii) ${\left(\frac{-13}{11}\right)}^2$
(iv) ${\left(\frac{1}{6}\right)}^3$
(v) ${\left(\frac{-1}{2}\right)}^5$
(vi) ${\left(\frac{-3}{2}\right)}^4$
(vii) ${\left(\frac{-4}{7}\right)}^3$
(viii) (−1)_⁹

Answer 1

(i) ${\left(\frac{2}{3}\right)}^5=\frac{{\left(2\right)}^5}{{\left(3\right)}^5}=\frac{2\times 2\times 2\times 2\times 2}{3\times 3\times 3\times 3\times 3}=\frac{32}{243}$

(ii) ${\left(\frac{-8}{5}\right)}^3=\frac{{\left(-8\right)}^3}{{\left(5\right)}^3}=\frac{\left(-8\right)\times \left(-8\right)\times \left(-8\right)}{5\times 5\times 5}=\frac{-512}{125}$

(iii) ${\left(\frac{-13}{11}\right)}^2=\frac{{\left(-13\right)}^2}{{\left(11\right)}^2}=\frac{\left(-13\right)\times \left(-13\right)}{11\times 11}=\frac{169}{121}$

(iv) ${\left(\frac{1}{6}\right)}^3=\frac{{\left(1\right)}^3}{{\left(6\right)}^3}=\frac{1\times 1\times 1}{6\times 6\times 6}=\frac{1}{216}$

(v) ${\left(\frac{-1}{2}\right)}^5=\frac{{\left(-1\right)}^5}{{\left(2\right)}^5}=\frac{\left(-1\right)\times \left(-1\right)\times \left(-1\right)\times \left(-1\right)\times \left(-1\right)}{2\times 2\times 2\times 2\times 2}=\frac{-1}{32}$

(vi) ${\left(\frac{-3}{2}\right)}^4=\frac{{\left(-3\right)}^4}{{\left(2\right)}^4}=\frac{\left(-3\right)\times \left(-3\right)\times \left(-3\right)\times \left(-3\right)}{2\times 2\times 2\times 2}=\frac{81}{16}$

(vii) ${\left(\frac{-4}{7}\right)}^3=\frac{{\left(-4\right)}^3}{{\left(7\right)}^3}=\frac{\left(-4\right)\times \left(-4\right)\times \left(-4\right)}{7\times 7\times 7}=\frac{-64}{343}$

(viii) ${\left(-1\right)}^9=-1$ [Since (-1) ^{an odd natural number} = -1]