Simplifying Expression: Simplify Expression

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Discussion Overview

The discussion revolves around simplifying a mathematical expression involving square roots and algebraic terms. Participants are exploring the simplification process and verifying the correctness of proposed solutions.

Discussion Character

  • Mathematical reasoning

Main Points Raised

  • One participant presents the expression to be simplified: $\dfrac{\sqrt{1+\sqrt{1-a^2}}((1+a)\sqrt{1+a}-(1-a)\sqrt{1-a})}{a(2+\sqrt{1-a^2})$.
  • Another participant proposes that the simplified result is $\sqrt{2}$.
  • A third participant agrees with the result of $\sqrt{2}$, stating it is the correct answer.
  • A later post reiterates the original expression for further simplification.

Areas of Agreement / Disagreement

There is a partial agreement on the result being $\sqrt{2}$, but it is unclear if all participants fully accept this conclusion, as the discussion includes a reiteration of the original expression.

Contextual Notes

The discussion does not clarify the assumptions or steps taken to arrive at the proposed simplifications, leaving potential gaps in the reasoning process.

anemone
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Simplify the expression $\dfrac{\sqrt{1+\sqrt{1-a^2}}((1+a)\sqrt{1+a}-(1-a)\sqrt{1-a})}{a(2+\sqrt{1-a^2})}$.
 
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$\sqrt 2$
Is it correct ?
 
Albert said:
$\sqrt 2$
Is it correct ?

Yes, $\sqrt{2}$ is the correct answer. :)
 
anemone said:
Simplify the expression $\dfrac{\sqrt{1+\sqrt{1-a^2}}((1+a)\sqrt{1+a}-(1-a)\sqrt{1-a})}{a(2+\sqrt{1-a^2})}---(1)$.
let :$x=\sqrt{1+a}, y=\sqrt{1-a}$
(1)becomes :$\dfrac{2\sqrt{1+xy}(x^3-y^3)}{(x^2-y^2)(2+xy)}$
$=\dfrac {2\sqrt{1+xy}}{x+y}=\dfrac{2}{\sqrt 2}=\sqrt 2$
for :$x^2+y^2=2, x^2-y^2=2a, (x+y)=\sqrt {2(1+xy)}$
 

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