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Simplify radicals in arithmetical expressions

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simplifyRadical(z) tries to simplify the radicals in the expression z. The result is mathematically equivalent to z.

radsimp and simplifyRadical are equivalent.


Example 1

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)))

Example 2

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


delete x, y, z:

Example 3

Use simplifyRadical to simplify these arithmetical expressions containing variables:

z := x/(sqrt(3) - 1) - x/2

simplifyRadical(z) = expand(radsimp(z))

delete z:

Example 4

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) * (4-2^(1/3))^(1/2)))



An arithmetical expression

Return Values

Arithmetical expression.


For constant algebraic expressions, simplifyRadical constructs a tower of algebraic extensions of 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. 169-188.In some special cases, an algorithm based on Landau, ”How to tangle with a nested radical”, The Mathematical Intelligencer 16, 1994, no. 2, pp. 49-55, is used.

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