How would one approach the following problems :
$\sqrt{\sqrt{5}-\sqrt{3-\sqrt{29-6\sqrt{20}}}}$
$(\frac12\sqrt{8}-3\sqrt{2}-\sqrt{10})*(\sqrt{2}+2\sqrt{1.6}-3\sqrt{0.4})$
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.
On
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
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.