<s>[INST] How to get the smallest positive integer solutions to $2\sqrt [3] {x} =3\sqrt [5] {y}$? How do I find the smallest positive integer solutions for $x$ and $y$ in the following equation? $$2\sqrt [3] {x} =3\sqrt [5] {y}$$
 [/INST] we have $$ 2\sqrt[3]{x} = 3\sqrt[5]{ y}$$ 
so cubing both side we have 
     $$2^3 x=3^3y^{\frac 35}$$
now 
      $$2^{ 3 \times 5}x^5=3^{ 3\times 5} y^3$$
so now we can write this 
      $$\frac{2^{ 3 \times 5}}{3^{ 3\times 5}}= \frac{y^3}{x^5}$$
so $$ y^3= (2^5)^3 $$
and  $$ x^5= (3^3)^5 $$
so$$ y=32$$ and $$x=27$$
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