If a² = log₁₀x, b³ = log₁₀y and $\dfrac{a^2}{2} - \dfrac{b^3}{3}$ = log₁₀z, express z in terms of x and y.

Given,

a² = log₁₀x

b³ = log₁₀y

Substituting above values in $\dfrac{a^2}{2} - \dfrac{b^3}{3} = \text{log}_{10}z$ we get,

$\Rightarrow \dfrac{\text{log}${10}x}{2} - \dfrac{\text{log}{10}y}{3} = \text{log}{10}z \\[1em] \Rightarrow \dfrac{3\text{log}{10}x - 2\text{log}{10}y}{6} = \text{log}{10}z \\[1em] \Rightarrow \dfrac{\text{log}{10}x^3 - \text{log}{10}y^2}{6} = \text{log}{10}z \\[1em] \Rightarrow \dfrac{1}{6}\text{log}{10}\dfrac{x^3}{y^2} = \text{log}{10}z \\[1em] \Rightarrow \text{log}{10}\Big(\dfrac{x^3}{y^2}\Big)^{\dfrac{1}{6}} = \text{log}{10}z \\[1em] \Rightarrow \text{log}{10}\Big(\dfrac{x^\dfrac{1}{2}}{y^\dfrac{1}{3}}\Big) = \text{log}{10}z \\[1em] \Rightarrow \text{log}{10}\dfrac{\sqrt{x}}{\sqrt[3]y} = \text{log}_{10}z \\[1em] \Rightarrow z = \dfrac{\sqrt{x}}{\sqrt[3]y}.

Hence, $z = \dfrac{\sqrt{x}}{\sqrt[3]y}.$