def test_sympyissue_7638():
f = pi/log(sqrt(2))
assert ((1 + I)**(I*f/2))**0.3 == (1 + I)**(0.15*I*f)
# if 1/3 -> 1.0/3 this should fail since it cannot be shown that the
# sign will be +/-1; for the previous "small arg" case, it didn't matter
# that this could not be proved
assert (1 + I)**(4*I*f) == ((1 + I)**(12*I*f))**Rational(1, 3)
assert (((1 + I)**(I*(1 + 7*f)))**Rational(1, 3)).exp == Rational(1, 3)
r = symbols('r', extended_real=True)
assert sqrt(r**2) == abs(r)
assert cbrt(r**3) != r
assert sqrt(Pow(2*I, 5*S.Half)) != (2*I)**(5/Integer(4))
p = symbols('p', positive=True)
assert cbrt(p**2) == p**(2/Integer(3))
assert NS(((0.2 + 0.7*I)**(0.7 + 1.0*I))**(0.5 - 0.1*I), 1) == '0.4 + 0.2*I'
assert sqrt(1/(1 + I)) == sqrt((1 - I)/2) # or 1/sqrt(1 + I)
e = 1/(1 - sqrt(2))
assert sqrt(e) == I/sqrt(-1 + sqrt(2))
assert e**-S.Half == -I*sqrt(-1 + sqrt(2))
assert sqrt((cos(1)**2 + sin(1)**2 - 1)**(3 + I)).exp == S.Half
assert sqrt(r**(4/Integer(3))) != r**(2/Integer(3))
assert sqrt((p + I)**(4/Integer(3))) == (p + I)**(2/Integer(3))
assert sqrt((p - p**2*I)**2) == p - p**2*I
assert sqrt((p + r*I)**2) != p + r*I
e = (1 + I/5)
assert sqrt(e**5) == e**(5*S.Half)
assert sqrt(e**6) == e**3
assert sqrt((1 + I*r)**6) != (1 + I*r)**3
def test_root():
from diofant.abc import x
n = Symbol('n', integer=True)
k = Symbol('k', integer=True)
assert root(2, 2) == sqrt(2)
assert root(2, 1) == 2
assert root(2, 3) == 2**Rational(1, 3)
assert root(2, 3) == cbrt(2)
assert root(2, -5) == 2**Rational(4, 5) / 2
assert root(-2, 1) == -2
assert root(-2, 2) == sqrt(2) * I
assert root(-2, 1) == -2
assert root(x, 2) == sqrt(x)
assert root(x, 1) == x
assert root(x, 3) == x**Rational(1, 3)
assert root(x, 3) == cbrt(x)
assert root(x, -5) == x**Rational(-1, 5)
assert root(x, n) == x**(1 / n)
assert root(x, -n) == x**(-1 / n)
assert root(x, n, k) == x**(1 / n) * (-1)**(2 * k / n)
def test_sympyissue_3449():
# test if powers are simplified correctly
# see also issue sympy/sympy#3995
assert cbrt(x)**2 == x**Rational(2, 3)
assert (x**3)**Rational(2, 5) == Pow(x**3, Rational(2, 5), evaluate=False)
a = Symbol('a', extended_real=True)
b = Symbol('b', extended_real=True)
assert (a**2)**b == (abs(a)**b)**2
assert sqrt(1/a) != 1/sqrt(a) # e.g. for a = -1
assert cbrt(a**3) != a
assert (x**a)**b != x**(a*b) # e.g. x = -1, a=2, b=1/2
assert (x**.5)**b == x**(.5*b)
assert (x**.5)**.5 == x**.25
assert (x**2.5)**.5 != x**1.25 # e.g. for x = 5*I
k = Symbol('k', integer=True)
m = Symbol('m', integer=True)
assert (x**k)**m == x**(k*m)
assert Number(5)**Rational(2, 3) == cbrt(Number(25))
assert (x**.5)**2 == x**1.0
assert (x**2)**k == (x**k)**2 == x**(2*k)
a = Symbol('a', positive=True)
assert (a**3)**Rational(2, 5) == a**Rational(6, 5)
assert (a**2)**b == (a**b)**2
assert (a**Rational(2, 3))**x == (a**(2*x/3)) != (a**x)**Rational(2, 3)
def test_sympyissue_3449():
# test if powers are simplified correctly
# see also issue sympy/sympy#3995
assert cbrt(x)**2 == x**Rational(2, 3)
assert (x**3)**Rational(2, 5) == Pow(x**3, Rational(2, 5), evaluate=False)
a = Symbol('a', extended_real=True)
b = Symbol('b', extended_real=True)
assert (a**2)**b == (abs(a)**b)**2
assert sqrt(1/a) != 1/sqrt(a) # e.g. for a = -1
assert cbrt(a**3) != a
assert (x**a)**b != x**(a*b) # e.g. x = -1, a=2, b=1/2
assert (x**.5)**b == x**(.5*b)
assert (x**.5)**.5 == x**.25
assert (x**2.5)**.5 != x**1.25 # e.g. for x = 5*I
k = Symbol('k', integer=True)
m = Symbol('m', integer=True)
assert (x**k)**m == x**(k*m)
assert Number(5)**Rational(2, 3) == cbrt(Number(25))
assert (x**.5)**2 == x**1.0
assert (x**2)**k == (x**k)**2 == x**(2*k)
a = Symbol('a', positive=True)
assert (a**3)**Rational(2, 5) == a**Rational(6, 5)
assert (a**2)**b == (a**b)**2
assert (a**Rational(2, 3))**x == (a**(2*x/3)) != (a**x)**Rational(2, 3)
def test_sympyissue_7638():
f = pi/log(sqrt(2))
assert ((1 + I)**(I*f/2))**0.3 == (1 + I)**(0.15*I*f)
# if 1/3 -> 1.0/3 this should fail since it cannot be shown that the
# sign will be +/-1; for the previous "small arg" case, it didn't matter
# that this could not be proved
assert (1 + I)**(4*I*f) == cbrt((1 + I)**(12*I*f))
assert cbrt((1 + I)**(I*(1 + 7*f))).exp == Rational(1, 3)
r = symbols('r', extended_real=True)
assert sqrt(r**2) == abs(r)
assert cbrt(r**3) != r
assert sqrt(Pow(2*I, Rational(5, 2))) != (2*I)**Rational(5, 4)
p = symbols('p', positive=True)
assert cbrt(p**2) == p**Rational(2, 3)
assert NS(((0.2 + 0.7*I)**(0.7 + 1.0*I))**(0.5 - 0.1*I), 1) == '0.4 + 0.2*I'
assert sqrt(1/(1 + I)) == sqrt((1 - I)/2) # or 1/sqrt(1 + I)
e = 1/(1 - sqrt(2))
assert sqrt(e) == I/sqrt(-1 + sqrt(2))
assert e**Rational(-1, 2) == -I*sqrt(-1 + sqrt(2))
assert sqrt((cos(1)**2 + sin(1)**2 - 1)**(3 + I)).exp == Rational(1, 2)
assert sqrt(r**Rational(4, 3)) != r**Rational(2, 3)
assert sqrt((p + I)**Rational(4, 3)) == (p + I)**Rational(2, 3)
assert sqrt((p - p**2*I)**2) == p - p**2*I
assert sqrt((p + r*I)**2) != p + r*I
e = (1 + I/5)
assert sqrt(e**5) == e**Rational(5, 2)
assert sqrt(e**6) == e**3
assert sqrt((1 + I*r)**6) != (1 + I*r)**3
def test_real_root():
assert real_root(-8, 3) == -2
assert real_root(-16, 4) == root(-16, 4)
r = root(-7, 4)
assert real_root(r) == r
r1 = root(-1, 3)
r2 = r1**2
r3 = root(-1, 4)
assert real_root(r1 + r2 + r3) == -1 + r2 + r3
assert real_root(root(-2, 3)) == -root(2, 3)
assert real_root(-8., 3) == -2
x = Symbol('x')
n = Symbol('n')
g = real_root(x, n)
assert g.subs(dict(x=-8, n=3)) == -2
assert g.subs(dict(x=8, n=3)) == 2
# give principle root if there is no real root -- if this is not desired
# then maybe a Root class is needed to raise an error instead
assert g.subs(dict(x=I, n=3)) == cbrt(I)
assert g.subs(dict(x=-8, n=2)) == sqrt(-8)
assert g.subs(dict(x=I, n=2)) == sqrt(I)
def test_large_rational():
e = cbrt(Rational(123712**12 - 1, 7) + Rational(1, 7))
assert e == 234232585392159195136*cbrt(Rational(1, 7))
def test_large_rational():
e = cbrt(Rational(123712**12 - 1, 7) + Rational(1, 7))
assert e == 234232585392159195136*cbrt(Rational(1, 7))