Exercice 8 Rendre rationnel le dénominateur de la fraction : 2−33+1\dfrac{2-\sqrt3}{\sqrt3+1}3+12−3 Calculer (1−2)2(1-\sqrt2)^2(1−2)2, puis déduire 3−22\sqrt{3-2\sqrt2}3−22 Correction (2−3)(3−1)(3+1)(3−1)=23−2−3+332−12=33−52\begin{align*} \dfrac{(2-\sqrt3){\bf\left(\color{red}\sqrt3-1\right)}}{(\sqrt3+1){\bf\left(\color{red}\sqrt3-1\right)}} &=\dfrac{2\sqrt3-2-3+\sqrt3}{\sqrt3^2-1^2} \\ &=\dfrac{3\sqrt3-5}{2} \end{align*}(3+1)(3−1)(2−3)(3−1)=32−1223−2−3+3=233−5 ∙ (1−2)2=12−2×1×2+22=3−23\begin{align*} \bullet~~(1-\sqrt2)^2 &=1^2-2\times1\times\sqrt2+\sqrt2^2 \\ &=3-2\sqrt3 \end{align*}∙ (1−2)2=12−2×1×2+22=3−23 ∙ 3−22=(1−2)2=−(1−2)car 1<2=2−1\begin{align*} \bullet~~\sqrt{3-2\sqrt2} &=\sqrt{(1-\sqrt2)^2}\\ &=-(1-\sqrt2) & \text{car } 1< \sqrt2 \\ &=\sqrt2-1 \end{align*}∙ 3−22=(1−2)2=−(1−2)=2−1car 1<2