Responder:
El valor es #-2#
Explicación:
Dejar # x = (2 + sqrt5) ^ (1/3) + (2-sqrt5) ^ (1/3) # entonces
# x ^ 3 = {(2 + sqrt5) ^ (1/3) + (2-sqrt5) ^ (1/3)} ^ 3 #
Recordatorio:
# (a + b) ^ 3 = a ^ 3 + b ^ 3 + 3ab (a + b), a ^ 2-b ^ 2 = (a + b) (a-b) #
y deja # a = (2 + sqrt5), b = (2-sqrt5):. ab = 4-5 = -1 #
#:. x ^ 3 = (2 + sqrt5) ^ (3 * 1/3) + (2-sqrt5) ^ (3 * 1/3) +3 (2 + sqrt5) (2-sqrt5) (2 + sqrt5 + 2-sqrt5) #
o # x ^ 3 = 2 + cancelar (sqrt5) + 2-cancelar (sqrt5) +3 (4-5) (2 + cancelar (sqrt5) + 2-cancelar (sqrt5)) #
o # x ^ 3 = 4 + 3 (-1) (4) o x ^ 3 = 4 -12 o x ^ 3 = -8 # o
#x = (-8) ^ (1/3) = -2 #. El valor es #-2# Respuesta
