Rationalize the denominator:
3−13+1\dfrac{\sqrt{3} - 1}{\sqrt{3} + 1}3+13−1.
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Rationalizing the denominator,
⇒3−13+1×3−13−1⇒(3−1)2(3)2−(1)2⇒(3)2+(1)2−2×3×13−1⇒3+1−232⇒(4−23)2⇒2(2−3)2⇒(2−3)\Rightarrow \dfrac{\sqrt{3} - 1}{\sqrt{3} + 1} \times \dfrac{\sqrt{3} - 1}{\sqrt{3} - 1} \\[1em] \Rightarrow \dfrac{(\sqrt{3} - 1)^2}{(\sqrt{3})^2 - (1)^2} \\[1em] \Rightarrow \dfrac{(\sqrt{3})^2 + (1)^2 - 2 \times \sqrt{3} \times 1 }{3 - 1} \\[1em] \Rightarrow \dfrac{3 + 1 - 2\sqrt{3}}{2} \\[1em] \Rightarrow \dfrac{(4 - 2\sqrt{3})}{2} \\[1em] \Rightarrow \dfrac{2(2 - \sqrt{3})}{2} \\[1em] \Rightarrow (2 - \sqrt{3})⇒3+13−1×3−13−1⇒(3)2−(1)2(3−1)2⇒3−1(3)2+(1)2−2×3×1⇒23+1−23⇒2(4−23)⇒22(2−3)⇒(2−3)
Hence, on rationalizing = 3−13+1=(2−3)\dfrac{\sqrt{3} - 1}{\sqrt{3} + 1} = (2 - \sqrt{3})3+13−1=(2−3).
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