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Simplify : 7310+3−256+5−3215+32\dfrac{7\sqrt{3}}{\sqrt{10} + \sqrt{3}} - \dfrac{2\sqrt{5}}{\sqrt{6} + \sqrt{5}} - \dfrac{3\sqrt{2}}{\sqrt{15} + 3\sqrt{2}}

Rational Irrational Nos

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Answer

7310+3−256+5−3215+32\dfrac{7\sqrt{3}}{\sqrt{10} + \sqrt{3}} - \dfrac{2\sqrt{5}}{\sqrt{6} + \sqrt{5}} - \dfrac{3\sqrt{2}}{\sqrt{15} + 3\sqrt{2}} ….(i)\qquad \text{….(i)}

Simplifying each term individually,

7310+3\dfrac{7\sqrt{3}}{\sqrt{10} + \sqrt{3}}

Let us rationalize its denominator,

7310+3=7310+3×10−310−3⇒(73)(10−3)(10)2−(3)2⇒73×10−73×3(10)2−(3)2⇒73×10−73×3(10)2−(3)2⇒730−(7×3)10−3⇒(730−(7×3)7)⇒7×(30−37)⇒30−3….(ii)\dfrac{7\sqrt{3}}{\sqrt{10} + \sqrt{3}} = \dfrac{7\sqrt{3}}{\sqrt{10} + \sqrt{3}} × \dfrac{\sqrt{10} - \sqrt{3}}{\sqrt{10} - \sqrt{3}} \\[1.5em] \Rightarrow\dfrac{(7\sqrt{3})(\sqrt{10} - \sqrt{3}) }{(\sqrt{10})^2 - (\sqrt{3})^2} \\[1.5em]\Rightarrow\dfrac{7\sqrt{3} × \sqrt{10} - 7\sqrt{3} × \sqrt{3} }{(\sqrt{10})^2 - (\sqrt{3})^2} \\[1.5em] \Rightarrow\dfrac{7\sqrt{3 × 10} - 7\sqrt{3 × 3} }{(\sqrt{10})^2 - (\sqrt{3})^2} \\[1.5em] \Rightarrow\dfrac{7\sqrt{30}-(7×3)}{10-3} \\[1.5em] \Rightarrow\Big(\dfrac{7\sqrt{30}-(7×3)}{7}\Big) \\[1.5em] \Rightarrow 7 ×\Big(\dfrac{\sqrt{30} - 3}{7}\Big) \\[1.5em] \bold{\Rightarrow{\sqrt{30}-3}} \qquad \text{….(ii)} \\[1.5em]

256+5\dfrac{2\sqrt{5}}{\sqrt{6} + \sqrt{5}}

Let us rationalize its denominator,

256+5=256+5×6−56−5⇒(25)(6−5)(6)2−(5)2⇒25×6−25×5(6)2−(5)2⇒25×6−25×56−5⇒230−101⇒230−10….(iii)\dfrac{2\sqrt{5}}{\sqrt{6} + \sqrt{5}} = \dfrac{2\sqrt{5}}{\sqrt{6} + \sqrt{5}} × \dfrac{\sqrt6 - \sqrt{5}}{\sqrt{6} - \sqrt{5}} \\[1.5em] \Rightarrow\dfrac{(2\sqrt{5})(\sqrt{6} - \sqrt{5}) }{(\sqrt{6})^2 - (\sqrt{5})^2} \\[1.5em] \Rightarrow\dfrac{2\sqrt{5} × \sqrt{6} - 2\sqrt{5} ×\sqrt{5} }{(\sqrt{6})^2 - (\sqrt{5})^2} \\[1.5em] \Rightarrow\dfrac{2\sqrt{5 × 6} - 2\sqrt{5 × 5} }{6 - 5} \\[1.5em] \Rightarrow\dfrac{2\sqrt{30} - 10}{1} \\[1.5em] \bold{\Rightarrow{2\sqrt{30} - 10}} \qquad \text{….(iii)} \\[1.5em]

3215+32\dfrac{3\sqrt{2}}{\sqrt{15} + 3\sqrt{2}}

Let us rationalize its denominator,

3215+32=3215+32×15−3215−32⇒(32)(15−32)(15)2−(32)2⇒32×15−32×32(15)2−(32)2⇒330−1815−18⇒330−18−3⇒[−3(−30+6)−3]⇒−(−30+61)⇒6−30….(iv)\dfrac{3\sqrt{2}}{\sqrt{15} + 3\sqrt{2}} = \dfrac{3\sqrt{2}}{\sqrt{15} + 3\sqrt{2}} × \dfrac{\sqrt{15} - 3\sqrt{2}}{\sqrt{15} - 3\sqrt{2}} \\[1.5em] \Rightarrow\dfrac{(3\sqrt{2})(\sqrt{15} - 3\sqrt{2}) }{(\sqrt{15})^2 - (3\sqrt{2})^2} \\[1.5em] \Rightarrow\dfrac{3\sqrt{2} × \sqrt{15} - 3\sqrt{2} × 3\sqrt{2} }{(\sqrt{15})^2 - (3\sqrt{2})^2} \\[1.5em] \Rightarrow\dfrac{3\sqrt{30} - 18}{15 - 18} \\[1.5em] \Rightarrow\dfrac{3\sqrt{30} - 18}{-3} \\[1.5em] \Rightarrow\Big[\dfrac{-3(-\sqrt{30} + 6)}{-3}\Big] \\[1.5em] \Rightarrow-\Big(\dfrac{-\sqrt{30} + 6}{1}\Big) \\[1.5em] \bold{\Rightarrow{6 - \sqrt{30}}} \qquad \text{….(iv)} \\[1.5em]

Using (ii) , (iii) , (iv) in equation (i):

7310+3−256+5−3215+32=(30−3)−(230−10)−(6−30)\dfrac{7\sqrt{3}}{\sqrt{10} + \sqrt{3}} - \dfrac{2\sqrt{5}}{\sqrt{6} + \sqrt{5}} - \dfrac{3\sqrt{2}}{\sqrt{15} + 3\sqrt{2}} =(\sqrt{30} - 3)-(2\sqrt{30} - 10) - (6 -\sqrt{30}) \\[1.5em] ⇒30−3−230+10−6+30\Rightarrow \sqrt{30} - 3 - 2\sqrt{30} + 10 - 6 + \sqrt{30} \\[1.5em] ⇒230−230−3+10−6\Rightarrow 2\sqrt{30} - 2\sqrt{30} -3 + 10 - 6 \\[1.5em] ⇒1\bold{\Rightarrow 1}

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