Simplify radicals in arithmetical expressions
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simplifyRadical(z
)
simplifyRadical(z)
tries to simplify the
radicals in the expression z
. The result is mathematically
equivalent to z
.
radsimp
and simplifyRadical
are
equivalent.
Simplify these constant expressions with square roots and higher order radicals:
simplifyRadical(3*sqrt(7)/(sqrt(7)  2)), simplifyRadical(sqrt(5 + 2*sqrt(6))); simplifyRadical(sqrt(5*sqrt(3) + 6*sqrt(2))), simplifyRadical(sqrt(3 + 2*sqrt(2)))
simplifyRadical((1/2 + 1/4*3^(1/2))^(1/2))
simplifyRadical((5^(1/3)  4^(1/3))^(1/2))
simplifyRadical(sqrt(3*sqrt(3 + 2*sqrt(5  12*sqrt(3  2*sqrt(2)))) + 14))
simplifyRadical(2*2^(1/4) + 2^(3/4)  (6*2^(1/2) + 8)^(1/2))
simplifyRadical(sqrt(1 + sqrt(3)) + sqrt(3 + 3*sqrt(3))  sqrt(10 + 6*sqrt(3)))
Create the following expression and then simplify it using simplifyRadical
:
x := sqrt(3)*I/2 + 1/2: y := x^(1/3) + x^(1/3): z := y^3  3*y
simplifyRadical(z)
delete x, y, z:
Use simplifyRadical
to simplify these arithmetical
expressions containing variables:
z := x/(sqrt(3)  1)  x/2
simplifyRadical(z) = expand(radsimp(z))
delete z:
Use simplifyRadical
to simplify nested radicals.
When simplifying nested radicals, simplifyRadical
tries
to reduce the nesting depth:
simplifyRadical((6*2^(1/2) + 8)^(1/2)); simplifyRadical(((32/5)^(1/5)  (27/5)^(1/5))^(1/3)); simplifyRadical(sqrt((3+2^(1/3))^(1/2) * (42^(1/3))^(1/2)))

Arithmetical expression.
For constant algebraic expressions, simplifyRadical
constructs
a tower of algebraic extensions of ℚ using
the domain Dom::AlgebraicExtension
.
It tries to return the simplest possible form.
This function is based on an algorithm described in Borodin, Fagin, Hopcroft and Tompa, "Decreasing the Nesting Depth of Expressions Involving Square Roots", JSC 1, 1985, pp. 169188.In some special cases, an algorithm based on Landau, "How to tangle with a nested radical", The Mathematical Intelligencer 16, 1994, no. 2, pp. 4955, is used.