How is $\sqrt{6}-\sqrt{2}$ equal to $2\sqrt{2-\sqrt{3}}$?

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Discussion Overview

The discussion centers on the mathematical expression $\sqrt{6}-\sqrt{2}$ and its potential equivalence to $2\sqrt{2-\sqrt{3}}$. Participants explore the validity of this claim, examining the expressions and their numerical approximations.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant asserts that $\sqrt{6}-\sqrt{2}$ does not equal $2\sqrt{2-\sqrt{3}}$, providing numerical approximations for both expressions.
  • Another participant reiterates the previous claim, emphasizing the correct interpretation of the expression as $2\sqrt{2-\sqrt{3}}$ rather than $2\sqrt{2}-\sqrt{3}$.
  • A later reply suggests a manipulation of the expression $\sqrt{6}-\sqrt{2}$, proposing that it can be expressed as $\sqrt{2}(\sqrt{3}-1)$ and further transformed to relate to $2\sqrt{2-\sqrt{3}}$.

Areas of Agreement / Disagreement

Participants generally disagree on the equivalence of the two expressions, with some asserting they are not equal while others explore potential manipulations that might suggest a relationship.

Contextual Notes

Participants rely on numerical approximations and algebraic manipulations, but the discussion does not resolve the equivalence or provide a definitive conclusion regarding the expressions.

Drain Brain
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how is $\sqrt{6}-\sqrt{2}$ equal to $2\sqrt{2-\sqrt{3}}$

please explain. Thanks!
 
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It does not. $\sqrt 6 - \sqrt 2 \approx 1.03528$, and $2 \sqrt 2 - \sqrt 3 \approx 1.09638$.
 
magneto said:
It does not. $\sqrt 6 - \sqrt 2 \approx 1.03528$, and $2 \sqrt 2 - \sqrt 3 \approx 1.09638$.

Notice OP wrote $2 \sqrt{2 - \sqrt{3}}$, not $2 \sqrt{2} - \sqrt{3}$.
 
Oops. The font rendering in Safari is messing up.

Then, $\sqrt 6 - \sqrt 2 = \sqrt 2 (\sqrt 3 - 1)$. Since the number are positive, use $a = \sqrt{a^2}$. $ \sqrt 2 (\sqrt 3 - 1) = \sqrt{2 (\sqrt{3}-1)^2}$, and deduce from there $2\sqrt{2-\sqrt 3}$.
 

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