[tex]\it \sqrt2\cdot\sqrt{2^2}\cdot\sqrt{2^3}\cdot\sqrt{2^4}=\sqrt{2\cdot2^2\cdot2^3\cdot2^4}=\sqrt{2^{1+2+3+4}}=\sqrt{2^{10}}=\\ \\ =\sqrt{(2^5)^2}=2^5=32=\dfrac{32}{1}\in\mathbb{Q}[/tex]
