An algebra problem by Prem Chebrolu

Algebra Level 3

$\text P+\sqrt{3}\text Q + \sqrt{5}\text R +\sqrt{15}\text S = \frac{1}{1+\sqrt{3}+\sqrt{5}}$ Then the value of $\text P$ . Where $\text{P, Q, R, S}$ are rational.

\frac{-1}{11}

\frac{3}{11}

\frac{7}{11}

\frac{-2}{11}

1 solution

Naren Bhandari
May 10, 2018

Let's us take right hand side $M=\dfrac{1}{1+\sqrt3 + \sqrt5} = \dfrac{1 +\sqrt 3 - \sqrt 5 }{\,(1 +\sqrt 3)^2 - \sqrt 5^2 } \phantom{ccccc}{\color{#3D99F6}\text{rationalization}}\\ M = \dfrac{1 +\sqrt 3 - \sqrt 5 }{\,(1 + 2\sqrt 3 + 3)- 5 } = \dfrac{ 1 + \sqrt 3 -\sqrt 5}{2\sqrt3 -1 } = \dfrac{ \,(1 +\sqrt 3 -\sqrt 5)\,(2\sqrt{3} +1) }{12-1} \\ M = \dfrac{2\sqrt 3 +1 + 6 + \sqrt 3 - 2\sqrt {15} -\sqrt 5 }{11} = \dfrac{7 + 3\sqrt 3 -\sqrt 5 -2 \sqrt{15} }{11} =\dfrac{P + Q\sqrt 3 - R\sqrt5 - S\sqrt {15}}{11}$ Hence, the value of $P = \boxed{\dfrac{7}{11}}$

An algebra problem by Prem Chebrolu

1 solution

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