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Question

Simplify (i) $\left(3-\sqrt{11}\right)\left(3+\sqrt{11}\right)\phantom{\rule{0ex}{0ex}}$ (ii) $\left(-3+\sqrt{5}\right)\left(-3-\sqrt{5}\right)$ (iii) ${\left(3-\sqrt{3}\right)}^{2}$ (iv) ${\left(\sqrt{5}-\sqrt{3}\right)}^{2}$ (v) $\left(5+\sqrt{7}\right)\left(2+\sqrt{5}\right)$ (vi) $\left(\sqrt{5}-\sqrt{2}\right)\left(\sqrt{2}-\sqrt{3}\right)$

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Solution

(i) $\left(3-\sqrt{11}\right)\left(3+\sqrt{11}\right)\phantom{\rule{0ex}{0ex}}$ $={3}^{2}-{\left(\sqrt{11}\right)}^{2}\left[\left(a-b\right)\left(a+b\right)={a}^{2}-{b}^{2}\right]\phantom{\rule{0ex}{0ex}}=9-11\phantom{\rule{0ex}{0ex}}=-2$ (ii) $\left(-3+\sqrt{5}\right)\left(-3-\sqrt{5}\right)$ $={\left(-3\right)}^{2}-{\left(\sqrt{5}\right)}^{2}\left[\left(a+b\right)\left(a-b\right)={a}^{2}-{b}^{2}\right]\phantom{\rule{0ex}{0ex}}=9-5\phantom{\rule{0ex}{0ex}}=4$ (iii) ${\left(3-\sqrt{3}\right)}^{2}$ $={3}^{2}+{\left(\sqrt{3}\right)}^{2}-2×3×\sqrt{3}\left[{\left(a-b\right)}^{2}={a}^{2}+{b}^{2}-2ab\right]\phantom{\rule{0ex}{0ex}}=9+3-6\sqrt{3}\phantom{\rule{0ex}{0ex}}=12-6\sqrt{3}$ (iv) ${\left(\sqrt{5}-\sqrt{3}\right)}^{2}$ $={\left(\sqrt{5}\right)}^{2}+{\left(\sqrt{3}\right)}^{2}-2×\sqrt{5}\sqrt{3}\left[{\left(a-b\right)}^{2}={a}^{2}+{b}^{2}-2ab\right]\phantom{\rule{0ex}{0ex}}=5+3-2\sqrt{15}=8-2\sqrt{15}$ $=\sqrt{5}×\sqrt{2}-\sqrt{5}×\sqrt{3}-\sqrt{2}×\sqrt{2}+\sqrt{2}×\sqrt{3}\phantom{\rule{0ex}{0ex}}=\sqrt{10}-\sqrt{15}-2+\sqrt{6}$ (v) $\left(5+\sqrt{7}\right)\left(2+\sqrt{5}\right)$ $\left(5+\sqrt{7}\right)\left(2+\sqrt{5}\right)\phantom{\rule{0ex}{0ex}}=5×2+5×\sqrt{5}+\sqrt{7}×2+\sqrt{7}×\sqrt{5}\phantom{\rule{0ex}{0ex}}=10+5\sqrt{5}+2\sqrt{7}+\sqrt{35}$ (vi) $\left(\sqrt{5}-\sqrt{2}\right)\left(\sqrt{2}-\sqrt{3}\right)$ $\left(\sqrt{5}-\sqrt{2}\right)\left(\sqrt{2}-\sqrt{3}\right)\phantom{\rule{0ex}{0ex}}=\sqrt{5}×\sqrt{2}-\sqrt{5}×\sqrt{3}-\sqrt{2}×\sqrt{2}+\sqrt{2}×\sqrt{3}\phantom{\rule{0ex}{0ex}}=\sqrt{10}-\sqrt{15}-2+\sqrt{6}$

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