Radical denesting or Dynesty???

$\color{#3D99F6}{\sqrt{\frac{3}{2}+\sqrt{\frac{3}{8}+\sqrt{\frac{3}{8^2\times2}\dotsm\;to\;\infty}}}}$ Can be written as $\color{#D61F06}{\frac{1}{2}\sqrt{6+\sqrt{6+\sqrt{6+\dotsm}}}}$ .Let $\color{#69047E}{\sqrt{6+\sqrt{6+\dotsm}}=x}$ .Then it can be written as: $\color{#20A900}{\sqrt{6+x}=x\\6+x=x^2\\x^2-x-6=0\\x^2+2x-3x-6=0\rightarrow x(x+2)-3(x+2)=0\rightarrow(x-3)(x+2)=0\rightarrow x=3\;or\;x=-2}$ As the nested radical is positive so discard the negative root.The radical in the question can be written as: $\color{#3D99F6}{\sqrt[3]{\frac{15}{8}+\frac{1}{2}\sqrt{6+\sqrt{6+\dotsm}}}\\=\sqrt[3]{\frac{15}{8}+\frac{1}{2}\times3}\\=\sqrt[3]{\frac{15}{8}+\frac{3}{2}}\\=\sqrt[3]{\frac{15+12}{8}}\\=\sqrt[3]{\frac{27}{8}}=\frac{3}{2}=\boxed{1.5}}$