CAT 2017 Quantitative Ability

${\log }_{6^2}{\mathit{x}}^2+{\log }_6{\mathit{y}}^{\frac{1}{2}}+3{\log }_{6^3}{\mathit{y}}^{\frac{1}{2}}\mathit{z}=6$
$\Rightarrow \frac{2\mathrm{logx}}{2\log 6}+\frac{1}{2}\frac{\mathrm{logy}}{\log 6}+\frac{{\mathrm{logy}}^{\frac{1}{2}}\mathit{z}}{\log 6}=6$
$\Rightarrow 2\mathrm{logx}+\mathrm{logy}+2\log \left({\mathit{y}}^{\frac{1}{2}}\mathit{z}\right)=6\log 6^2$
  $\Rightarrow {\mathrm{logx}}^2+\mathrm{logy}+{\mathrm{logyz}}^2=\log 6^{12}$
     $\Rightarrow \log \left({\mathit{x}}^2\mathit{y}\cdot \mathit{y}\cdot {\mathit{z}}^2\right)=\log 6^{12}$
         $\Rightarrow {\left(\mathit{x}\mathit{y}\mathit{z}\right)}^2={\left(6^6\right)}^2$
           $\Rightarrow \mathit{x}\mathit{y}\mathit{z}=6^6$
Given x, y, z is in G.P Let x = a, y = ab, z = ${\mathit{a}\mathit{b}}^2$
$\Rightarrow \mathit{x}\mathit{y}\mathit{z}={\mathit{a}}^3{\mathit{b}}^3={\left(\mathit{a}\mathit{b}\right)}^3$
$\Rightarrow {\left(\mathit{a}\mathit{b}\right)}^3={\left(6^2\right)}^3$
Possible values of (a, b) satisfying the equation
(1,36) (2, 18) (3. 12). (4, 9). (9, 4), (12. 3). (18. 2). (36, 1)
Given, y - x is a perfect cube
=>ab - a is perfect cube
⇒a(b - 1) is perfect cube
Only possible when (a, b) =(9, 4)
∴ x = 9, y = 36, z= 144
∴ x + y + z = 9 + 36 + 144= 189