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[parent] square root of square root binomial (Topic)

Some people call the expressions of the form $ a\!+\!b\sqrt{c}$ the square root binomials, especially when $ c$ is an square-free integer greater than 1 (and $ a$ and $ b$ rational numbers). On the high school level one may learn to perform arithmetic operations between such binomials (see e.g. division), or also polynomials containing several square root terms. Taking the square root of a square root binomial is more difficult and usually results nested square roots. However, there are some exceptions if the numbers are appropriate. We have the formulae

$\displaystyle \sqrt{a+\sqrt{b}} = \sqrt{\frac{a+\sqrt{a^2-b}}{2}}+\sqrt{\frac{a-\sqrt{a^2-b}}{2}}$
and
$\displaystyle \sqrt{a-\sqrt{b}} = \sqrt{\frac{a+\sqrt{a^2-b}}{2}}-\sqrt{\frac{a-\sqrt{a^2-b}}{2}}.$
If $ a^2\!-\!b$ happens to be square of a rational number, then the formulae allow to convert the square roots on the left side into expressions without nested square roots.

For example, because $ 6^2\!-\!20 = 16 = 4^2$, we obtain

$\displaystyle \sqrt{6\!+\!2\sqrt{5}} = \sqrt{6\!+\!\sqrt{20}} = \sqrt{\frac{6\!+\!4}{2}}+\sqrt{\frac{6\!-\!4}{2}} = 1\!+\!\sqrt{5},$
and because $ 4^2\!-\!7 = 9 = 3^2$, we get
$\displaystyle \sqrt{4\!-\!\sqrt{7}} = \sqrt{\frac{4\!+\!3}{2}}-\sqrt{\frac{4\!-\!3}{2}} = \frac{\sqrt{7}\!-\!1}{\sqrt{2}} = \frac{\sqrt{14}\!-\!\sqrt{2}}{2}.$

Bibliography

1
K. V¨AISÄLÄ: Algebran oppi- ja esimerkkikirja I. - Werner Söderström osakeyhtiö, Porvoo & Helsinki (1952).



"square root of square root binomial" is owned by pahio.
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See Also: taking square root algebraically

Also defines:  square root binomial, square root polynomial
Keywords:  nested square roots

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proof of square root of square root binomial (Proof) by rspuzio
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Cross-references: rational number, square, numbers, square root, polynomials, division, rational numbers, integer, square-free, expressions
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This is version 7 of square root of square root binomial, born on 2005-06-21, modified 2005-06-30.
Object id is 7178, canonical name is SquareRootOfSquareRootBinomial.
Accessed 7331 times total.

Classification:
AMS MSC: 11A25 (Number theory :: Elementary number theory :: Arithmetic functions; related numbers; inversion formulas)
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Proof by Doug_K on 2007-12-24 09:02:29
This was very helpful...

Could anybody show a proof of how you would derive this formula? I would really appreciate it.

Thanks
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