test_power = ->
  run_test [

    # according to the notorious
    # "PEMDAS" mnemonic, the power
    # operator has the most precedence.
    # Strangely, Mathematica parses
    # a^1/2 as sqrt(a), not following PEMDAS,
    # probably because of some fancy
    # euristics, since, contrarily to the
    # case above, it also parses
    # a^b/2 as (a^b)/2.
    # I think this more standard/uniform handling
    # a-la-sympy is ok.
    "a^1/2 + a^1/2",
    "a",

    "2^(1/2)",
    "2^(1/2)",

    "2^(3/2)",
    "2*2^(1/2)",

    "(-2)^(1/2)",
    "i*2^(1/2)",

    "3^(4/3)",
    "3*3^(1/3)",

    "3^(-4/3)",
    "1/(3*3^(1/3))",

    "3^(5/3)",
    "3*3^(2/3)",

    "3^(2/3)-9^(1/3)",
    "0",

    "3^(10/3)",
    "27*3^(1/3)",

    "3^(-10/3)",
    "1/(27*3^(1/3))",

    "(1/3)^(10/3)",
    "1/(27*3^(1/3))",

    "(1/3)^(-10/3)",
    "27*3^(1/3)",

    "27^(2/3)",
    "9",

    "27^(-2/3)",
    "1/9",

    "102^(1/2)",
    "2^(1/2)*3^(1/2)*17^(1/2)",

    "32^(1/3)",
    "2*2^(2/3)",

    "9999^(1/2)",
    "3*11^(1/2)*101^(1/2)",

    "8^(1/2)",
    "2*2^(1/2)",

    "10000^(1/3)",
    "10*2^(1/3)*5^(1/3)",

    # we could take out a "18" from the radix but
    # we only handle this for small numbers in
    # "quickfactor" routine. TODO
    "8204861575751304355842204^(1/2)",
    "8204861575751304355842204^(1/2)",

    # see above
    "simplify(8204861575751304355842204^(1/2))",
    "8204861575751304355842204^(1/2)",

    "sqrt(-1/2 -1/2 * x)",
    "(-1/2*x-1/2)^(1/2)",

    "sqrt(x*y)",
    "(x*y)^(1/2)",

    "sqrt(1/x)",
    "(1/x)^(1/2)",

    "sqrt(x^y)",
    "(x^y)^(1/2)",

    "sqrt(x)^2",
    "x",

    "sqrt(x^2)",
    "abs(x)",

    # always true, whether x is real or not
     "sqrt(x^2)^2",
     "x^2",

    "3^(1/2)*i/9",
    "1/9*i*3^(1/2)",

    "(-4.0)^(1.5)",
    "-8.0*i",

    "(-4.0)^(3/2)",
    "-8.0*i",

    # usually the rectangular form is returned.
    "(-1)^(1/3)",
    #"(-1)^(1/3)",
    "1/2+1/2*i*3^(1/2)",

    # note how the "double" type
    # is toxic i.e. it propagates through
    # everything it touches.
    # also, that -0.500000... _really_ is 0.5
    # however we get some error in the calculations
    # so it doesn't end up being exactly equal to -0.5
    "(-1.0)^(2/3)",
    "-0.500000...+0.866025...*i",

    # this also has a nested radical
    # form but we are not calculating
    # that.
    "(-1)^(1/3)*2^(1/4)",
    #"(-1)^(1/3)*2^(1/4)",
    "1/2*2^(1/4)+1/2*i*2^(1/4)*3^(1/2)",

    "(-1)^(1/2)",
    "i",

    "sqrt(1000000)",
    "1000",

    "sqrt(-1000000)",
    "1000*i",

    "sqrt(2^60)",
    "1073741824",

    # this is why we factor irrationals

    "6^(1/3) 3^(2/3)",
    "3*2^(1/3)",

    # inverse of complex numbers

    "1/(2+3*i)",
    "2/13-3/13*i",

    "1/(2+3*i)^2",
    "-5/169-12/169*i",

    "(-1+3i)/(2-i)",
    "-1+i",

    # other

    "(0.0)^(0.0)",
    "1.0",

    "(-4.0)^(0.5)",
    "2.0*i",

    "(-4.0)^(-0.5)",
    "-0.5*i",

    "(-4.0)^(-1.5)",
    "0.125*i",

    # more complex number cases

    "(1+i)^2",
    "2*i",

    "(1+i)^(-2)",
    "-1/2*i",

    "(1+i)^(1/2)",
    #"(-1)^(1/8)*2^(1/4)",
    "i*2^(1/4)*sin(1/8*pi)+2^(1/4)*cos(1/8*pi)",

    "(1+i)^(-1/2)",
    "-(-1)^(7/8)/(2^(1/4))",

    "(1+i)^(0.5)",
    "1.098684...+0.455090...*i",

    "(1+i)^(-0.5)",
    "0.776887...-0.321797...*i",

    # test cases for simplification of polar forms, counterclockwise

    "exp(i*pi/2)",
    "i",

    "exp(i*pi)",
    "-1",

    "exp(i*3*pi/2)",
    "-i",

    "exp(i*2*pi)",
    "1",

    "exp(i*5*pi/2)",
    "i",

    "exp(i*3*pi)",
    "-1",

    "exp(i*7*pi/2)",
    "-i",

    "exp(i*4*pi)",
    "1",

    "exp(A+i*pi/2)",
    "i*exp(A)",

    "exp(A+i*pi)",
    "-exp(A)",

    "exp(A+i*3*pi/2)",
    "-i*exp(A)",

    "exp(A+i*2*pi)",
    "exp(A)",

    "exp(A+i*5*pi/2)",
    "i*exp(A)",

    "exp(A+i*3*pi)",
    "-exp(A)",

    "exp(A+i*7*pi/2)",
    "-i*exp(A)",

    "exp(A+i*4*pi)",
    "exp(A)",

    # test cases for simplification of polar forms, clockwise

    "exp(-i*pi/2)",
    "-i",

    "exp(-i*pi)",
    "-1",

    "exp(-i*3*pi/2)",
    "i",

    "exp(-i*2*pi)",
    "1",

    "exp(-i*5*pi/2)",
    "-i",

    "exp(-i*3*pi)",
    "-1",

    "exp(-i*7*pi/2)",
    "i",

    "exp(-i*4*pi)",
    "1",

    "exp(A-i*pi/2)",
    "-i*exp(A)",

    "exp(A-i*pi)",
    "-exp(A)",

    "exp(A-i*3*pi/2)",
    "i*exp(A)",

    "exp(A-i*2*pi)",
    "exp(A)",

    "exp(A-i*5*pi/2)",
    "-i*exp(A)",

    "exp(A-i*3*pi)",
    "-exp(A)",

    "exp(A-i*7*pi/2)",
    "i*exp(A)",

    "exp(A-i*4*pi)",
    "exp(A)",
  ]