Question

Find the reciprocal of the rational number ${\displaystyle {\left(\frac{1}{2}\right)}^2÷{\left(\frac{2}{3}\right)}^3}$

Answer 1

Given, ${\displaystyle {\left(\frac{1}{2}\right)}^2÷{\left(\frac{2}{3}\right)}^3=\frac{{\left(\frac{1}{2}\right)}^2}{{\left(\frac{2}{3}\right)}^3}}$  ......${\displaystyle [\because a÷b=\frac{a}{b}]}$

= ${\displaystyle \frac{\frac{{\left(1\right)}^2}{{\left(2\right)}^2}}{\frac{{\left(2\right)}^3}{{\left(3\right)}^3}}}$ = ${\displaystyle \frac{\left(\frac{1}{4}\right)}{\left(\frac{8}{27}\right)}}$  ......${\displaystyle [\because {\left(\frac{a}{b}\right)}^n=\frac{a^n}{b^n}]}$ [∵ 1² = 1, 2² = 4, 2³ = 8 and 3³ = 27]

= ${\displaystyle \frac{1}{4}\times \frac{27}{8}=\frac{27}{4\times 8}=\frac{27}{32}}$  ......${\displaystyle [\because \frac{a}{b}÷\frac{c}{d}=\frac{a}{b}\times \frac{d}{c}]}$

We know that, reciprocal of a rational number is obtained by interchanging the numerator and denominator

∴ Reciprocal og given number = ${\displaystyle \frac{32}{27}}$.