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How would one approach the following problems :

  1. $\sqrt{\sqrt{5}-\sqrt{3-\sqrt{29-6\sqrt{20}}}}$

  2. $(\frac12\sqrt{8}-3\sqrt{2}-\sqrt{10})*(\sqrt{2}+2\sqrt{1.6}-3\sqrt{0.4})$

I am not looking for actual solutions but for tips on how these type of expressions are solved. I get completely different results from the ones listed in the answers.

Martin Spasov
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    You should be trying to use the method of completing the square all the way. Otherwise, go for the calculator. You're lucky that for the problem in $(1)$, we have, $29-6\sqrt{20}=(\sqrt{20}-3)^2$. Then, you just use $\sqrt{x^2}=|x|$. Repeat till the answer is simplified / you can't proceed further. – learner Dec 28 '15 at 17:36
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    As for $(2)$, you have, $$\sqrt{1.6}=\sqrt{\dfrac{16}{10}}=\frac{4}{\sqrt{10}}\~\ \sqrt{0.4}=\sqrt{\dfrac{4}{10}}=\frac{2}{\sqrt{10}}$$

    Now simplify.

     – learner Dec 28 '15 at 17:43

2 Answers2

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HINT: The best way to proceed is completing the square and expressing the expression under the square root as a perfect square.

As for example in (1), $$\sqrt{29-6\sqrt{20}}=\sqrt{(\sqrt{20}-3)^2}=\sqrt{20}-3$$

And so on, you go.

SchrodingersCat
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Your example is nice:

$$\sqrt{\sqrt{5}-\sqrt{3-\sqrt{29-6\sqrt{20}}}} = 1$$

Was using a CAS implementation in Prolog (to be published in a few weeks):

Jekejeke Prolog 2, Runtime Library 1.2.2
(c) 1985-2017, XLOG Technologies GmbH, Switzerland
?- use_module(library(groebner/generic)).
% 21 consults and 0 unloads in 843 ms.
Yes
?- X is sqrt(sqrt(5)-sqrt(3-sqrt(29-6*sqrt(20)))).
X is 1

The implementation has picked up ideas from here:

Borodin, A., Fagin, R., Hopcroft, J.E. and Tompa, M. (1985): Decreasing the Nesting Depth of Expressions Involving Square Roots, J . Symbolic Computation (1985) 1, 169-188 http://www.sciencedirect.com/science/article/pii/S0747717185800134