Activités numériques II : Exercice 19 première année secondaire

$$\begin{align*} X&=\dfrac{(\sqrt{2})^{18}\times 12^{-3}}{24\times(\sqrt{3})^{-4}}\\ &=\dfrac{\left({(\sqrt{2})^{2}}\right)^9\times 12^{-3}}{2\times12\times\left({(\sqrt{3})^2}\right)^{-2}}\\ &=\dfrac{2^9\times12^{-3}}{2\times12\times3^{-2}}\\ &=\dfrac{2^9\times3^2}{2\times12\times12^3}\\ &=\dfrac{2^9\times3^2}{2\times12^4}=\dfrac{2^8\times3^2}{(2^2\times3)^4}\\ &=\dfrac{2^8\times3^2}{2^8\times3^4}=\dfrac{1}{3^2}=\dfrac{1}{9}.\\ \\ Y&=\sqrt{\dfrac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}}\\ &=\sqrt{\dfrac{(\sqrt{3}+\sqrt{2})^2}{(\sqrt{3}-\sqrt{2})(\sqrt{3}+\sqrt{2})}}\\ &=\sqrt{\dfrac{(\sqrt{3}+\sqrt{2})^2}{\sqrt{3}^2-\sqrt{2}^2}}\\ &=\sqrt{\dfrac{(\sqrt{3}+\sqrt{2})^2}{3-2}}\\ &=\sqrt{(\sqrt{3}+\sqrt{2})^2}\\ &=\left|{\sqrt{3}+\sqrt{2}}\right|=\sqrt{3}+\sqrt{2} \end{align*}$$