Find the triplicate ratio of :

(i) 3 : 4

(ii) $\dfrac{1}{2} : \dfrac{1}{3}$

(iii) 1³ : 2³

(i) The triplicate ratio of 3 : 4 is,

= 3³ : 4³ = 27 : 64

Hence, the triplicate ratio is 27 : 64.

(ii) The triplicate ratio of $\dfrac{1}{2} : \dfrac{1}{3}$ is,

$=\big(\dfrac{1}{2}\big)^3 : \big(\dfrac{1}{3}\big)^3 \\[0.5em] = \big(\dfrac{1}{8}\big) : \big(\dfrac{1}{27}\big) \\[0.5em] = \dfrac{\dfrac{1}{8}}{\dfrac{1}{27}} \\[0.5em] = \dfrac{27}{8} = 27 : 8.$

Hence, the triplicate ratio is 27 : 8.

(iii) The triplicate ratio of 1³ : 2³ is,

= (1³)³ : (2³)³ = 1⁹ : 2⁹ = 1: 512

Hence, the triplicate ratio is 1 : 512.