Let the geometric mean between \(6a^3b\) and \(24ab^3\) be \(p\). \[ \therefore \frac{6a^3b}{p} = \frac{p}{24ab^3} \Rightarrow p^2 = 6a^3b \times 24ab^3 \Rightarrow p^2 = 144a^4b^4 \Rightarrow p = \sqrt{144a^4b^4} = 12a^2b^2 \] \[ \therefore \text{The geometric mean between } 6a^3b \text{ and } 24ab^3 \text{ is } 12a^2b^2 \]