$\min (a+b+c)^6 = ?$

By AM-GM inequality :

$\begin{aligned} \frac a3 + \frac a3 + \frac a3 + \frac b2 + \frac b2 + c & \le 6 \sqrt [6] {\frac {\color{#3D99F6}a^3b^2c}{3^3 \times 2^2}} = \frac 6{\sqrt[6]{108}} & \small \color{#3D99F6} \text{Since }a^3b^2c = 1 \\ \implies (a+b+c)^6 & \ge \frac {6^6}{108} = \boxed{432} \end{aligned}$

Equality occurs when $a = \frac 3{\sqrt[6]{108}}$ , $b = \frac 2{\sqrt[6]{108}}$ , and $c = \frac 1{\sqrt[6]{108}}$ .

By AM-GM inequality, $\begin{aligned}(a+b+c)^6&=(\frac{a}{3}+\frac{a}{3}+\frac{a}{3}+\frac{b}{2}+\frac{b}{2}+c)^6\\&\ge\frac{6^6}{108}\\&=432\end{aligned}$

$\left\{432,\left\{a\to \frac{\sqrt{3}}{\sqrt[3]{2}},b\to \frac{2^{2/3}}{\sqrt{3}},c\to \frac{1}{\sqrt[3]{2} \sqrt{3}}\right\}\right\}$

OK, a few more details:

Defining $c$ in terms of $a$ and $b$ is: $c=\frac{1}{a^3\,b^2}$ .

Expanding $(a+b+\frac{1}{a^3\,b^2})^6$ gives $\frac{1}{a^{18} b^{12}}+\frac{6}{a^{15} b^9}+\frac{6}{a^{14} b^{10}}+\frac{15}{a^{12} b^6}+\frac{30}{a^{11} b^7}+\frac{15}{a^{10} b^8}+\frac{20}{a^9 b^3}+\frac{60}{a^8 b^4}+\frac{60}{a^7 b^5}+\frac{20}{a^6 b^6}+a^6+\frac{15}{a^6}+6 a^5 b+\frac{60}{a^5 b}+15 a^4 b^2+\frac{90}{a^4 b^2}+20 a^3 b^3+\frac{6 b^3}{a^3}+\frac{60}{a^3 b^3}+15 a^2 b^4+\frac{15}{a^2 b^4}+\frac{6 a^2}{b^2}+\frac{30 b^2}{a^2}+6 a b^5+\frac{30 a}{b}+\frac{60 b}{a}+b^6+60$

The derivative of that expression with respect to $a$ is $-\frac{18}{a^{19} b^{12}}-\frac{90}{a^{16} b^9}-\frac{84}{a^{15} b^{10}}-\frac{180}{a^{13} b^6}-\frac{330}{a^{12} b^7}-\frac{150}{a^{11} b^8}-\frac{180}{a^{10} b^3}-\frac{480}{a^9 b^4}-\frac{420}{a^8 b^5}-\frac{120}{a^7 b^6}-\frac{90}{a^7}-\frac{300}{a^6 b}-\frac{360}{a^5 b^2}+6 a^5-\frac{18 b^3}{a^4}-\frac{180}{a^4 b^3}+30 a^4 b-\frac{30}{a^3 b^4}+60 a^3 b^2-\frac{60 b^2}{a^3}+60 a^2 b^3-\frac{60 b}{a^2}+30 a b^4+\frac{12 a}{b^2}+6 b^5+\frac{30}{b}$

The derivative of that expression with respect to $b$ is $-\frac{12}{a^{18} b^{13}}-\frac{54}{a^{15} b^{10}}-\frac{60}{a^{14} b^{11}}-\frac{90}{a^{12} b^7}-\frac{210}{a^{11} b^8}-\frac{120}{a^{10} b^9}-\frac{60}{a^9 b^4}-\frac{240}{a^8 b^5}-\frac{300}{a^7 b^6}-\frac{120}{a^6 b^7}-\frac{60}{a^5 b^2}+6 a^5-\frac{180}{a^4 b^3}+30 a^4 b-\frac{180}{a^3 b^4}+60 a^3 b^2+\frac{18 b^2}{a^3}-\frac{60}{a^2 b^5}+60 a^2 b^3-\frac{12 a^2}{b^3}+\frac{60 b}{a^2}+30 a b^4-\frac{30 a}{b^2}+\frac{60}{a}+6 b^5$

Equating the last two expression to $0$ and solving the last two equations together gives a number of solutions and also the value of $c$ was recreated: $\left( \begin{array}{ccc} a & \frac{1}{3} \left(\frac{\sqrt[3]{2} a^5}{\sqrt[3]{-2 a^{12}-27 a^6+3 \sqrt{3} \sqrt{4 a^{18}+27 a^{12}}}}+\frac{\sqrt[3]{-2 a^{12}-27 a^6+3 \sqrt{3} \sqrt{4 a^{18}+27 a^{12}}}}{\sqrt[3]{2} a^3}-a\right) & \frac{9}{a^3 \left(\frac{\sqrt[3]{2} a^5}{\sqrt[3]{-2 a^{12}-27 a^6+3 \sqrt{3} \sqrt{4 a^{18}+27 a^{12}}}}+\frac{\sqrt[3]{-2 a^{12}-27 a^6+3 \sqrt{3} \sqrt{4 a^{18}+27 a^{12}}}}{\sqrt[3]{2} a^3}-a\right)^2} \\ a & -\frac{\left(1+i \sqrt{3}\right) a^5}{3\ 2^{2/3} \sqrt[3]{-2 a^{12}-27 a^6+3 \sqrt{3} \sqrt{4 a^{18}+27 a^{12}}}}-\frac{\left(1-i \sqrt{3}\right) \sqrt[3]{-2 a^{12}-27 a^6+3 \sqrt{3} \sqrt{4 a^{18}+27 a^{12}}}}{6 \sqrt[3]{2} a^3}-\frac{a}{3} & \frac{1}{a^3 \left(-\frac{\left(1+i \sqrt{3}\right) a^5}{3\ 2^{2/3} \sqrt[3]{-2 a^{12}-27 a^6+3 \sqrt{3} \sqrt{4 a^{18}+27 a^{12}}}}-\frac{\left(1-i \sqrt{3}\right) \sqrt[3]{-2 a^{12}-27 a^6+3 \sqrt{3} \sqrt{4 a^{18}+27 a^{12}}}}{6 \sqrt[3]{2} a^3}-\frac{a}{3}\right)^2} \\ a & -\frac{\left(1-i \sqrt{3}\right) a^5}{3\ 2^{2/3} \sqrt[3]{-2 a^{12}-27 a^6+3 \sqrt{3} \sqrt{4 a^{18}+27 a^{12}}}}-\frac{\left(1+i \sqrt{3}\right) \sqrt[3]{-2 a^{12}-27 a^6+3 \sqrt{3} \sqrt{4 a^{18}+27 a^{12}}}}{6 \sqrt[3]{2} a^3}-\frac{a}{3} & \frac{1}{a^3 \left(-\frac{\left(1-i \sqrt{3}\right) a^5}{3\ 2^{2/3} \sqrt[3]{-2 a^{12}-27 a^6+3 \sqrt{3} \sqrt{4 a^{18}+27 a^{12}}}}-\frac{\left(1+i \sqrt{3}\right) \sqrt[3]{-2 a^{12}-27 a^6+3 \sqrt{3} \sqrt{4 a^{18}+27 a^{12}}}}{6 \sqrt[3]{2} a^3}-\frac{a}{3}\right)^2} \\ -\sqrt[3]{-\frac{1}{2}} \sqrt{3} & -\frac{\sqrt[3]{-1} 2^{2/3}}{\sqrt{3}} & -\frac{\sqrt[3]{-\frac{1}{2}}}{\sqrt{3}} \\ \sqrt[3]{-\frac{1}{2}} \sqrt{3} & \frac{\sqrt[3]{-1} 2^{2/3}}{\sqrt{3}} & \frac{\sqrt[3]{-\frac{1}{2}}}{\sqrt{3}} \\ -\frac{\sqrt{3}}{\sqrt[3]{2}} & -\frac{2^{2/3}}{\sqrt{3}} & -\frac{1}{\sqrt[3]{2} \sqrt{3}} \\ \frac{\sqrt{3}}{\sqrt[3]{2}} & \frac{2^{2/3}}{\sqrt{3}} & \frac{1}{\sqrt[3]{2} \sqrt{3}} \\ -\frac{(-1)^{2/3} \sqrt{3}}{\sqrt[3]{2}} & -\frac{(-2)^{2/3}}{\sqrt{3}} & -\frac{(-1)^{2/3}}{\sqrt[3]{2} \sqrt{3}} \\ \frac{(-1)^{2/3} \sqrt{3}}{\sqrt[3]{2}} & \frac{(-2)^{2/3}}{\sqrt{3}} & \frac{(-1)^{2/3}}{\sqrt[3]{2} \sqrt{3}} \\ \end{array} \right)$

Only one of these solutions is numeric, real and greater than zero in all three elements of the solution triple. Evaluating and simplifying the expression to be minimized with those values gives: $(a+b+c)^6\text{/.}\, \left\{a\to \frac{\sqrt{3}}{\sqrt[3]{2}},b\to \frac{2^{2/3}}{\sqrt{3}},c\to \frac{1}{\sqrt[3]{2} \sqrt{3}}\right\} \Rightarrow 432$