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Mathematics

Rationalize the denominator:

1(1+5+3)\dfrac{1}{(1 + \sqrt{5} + \sqrt{3})}.

Rational Irrational Nos

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Answer

Rationalizing the denominator,

⇒1(1+5+3)×(1+5−3)(1+5−3)⇒(1+5−3)(1+5+3)×(1+5−3)⇒(1+5−3)(1+5)2−(3)2⇒(1+5−3)1+5+25−3⇒(1+5−3)(3+25)\Rightarrow \dfrac{1}{(1 + \sqrt{5} + \sqrt{3})} \times \dfrac{(1 + \sqrt{5} - \sqrt{3})} {(1 + \sqrt{5} - \sqrt{3})} \\[1em] \Rightarrow \dfrac{(1 + \sqrt{5} - \sqrt{3})}{(1 + \sqrt{5} + \sqrt{3}) \times (1 + \sqrt{5} - \sqrt{3})} \\[1em] \Rightarrow \dfrac{(1 + \sqrt{5} - \sqrt{3})}{(1 + \sqrt{5})^2 - (\sqrt{3})^2} \\[1em] \Rightarrow \dfrac{(1 + \sqrt{5} - \sqrt{3})}{1 + 5 + 2\sqrt{5} - 3} \\[1em] \Rightarrow \dfrac{(1 + \sqrt{5} - \sqrt{3})}{(3 + 2\sqrt{5})}

Rationalizing the denominator again,

⇒(1+5−3)(3+25)×(3−25)(3−25)⇒3−25+35−10−33+215(3)2−(25)2⇒−7+5−33+2159−20⇒−(7−5+33−215)−11⇒(7−5+33−215)11\Rightarrow \dfrac{(1 + \sqrt{5} - \sqrt{3})}{(3 + 2\sqrt{5})} \times \dfrac {(3 - 2\sqrt{5})}{(3 - 2\sqrt{5})} \\[1em] \Rightarrow \dfrac{3 - 2\sqrt{5} + 3\sqrt{5} - 10 - 3\sqrt{3} + 2\sqrt{15}}{(3)^2 - (2\sqrt{5})^2} \\[1em] \Rightarrow \dfrac{-7 + \sqrt{5} - 3\sqrt{3} + 2\sqrt{15}}{9 - 20} \\[1em] \Rightarrow \dfrac{-(7 - \sqrt{5} + 3\sqrt{3} - 2\sqrt{15})}{-11} \\[1em] \Rightarrow \dfrac{(7 - \sqrt{5} + 3\sqrt{3} - 2\sqrt{15})}{11}

Hence, on rationalizing 1(1+5+3)=(7−5+33−215)11\dfrac{1}{(1 + \sqrt{5} + \sqrt{3})} = \dfrac{(7 - \sqrt{5} + 3\sqrt{3} - 2\sqrt{15})} {11}.

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