Svar:
# A ^ 3 / b ^ 3 + b ^ 3 / a ^ 3 = 0or10sqrt13 #
Forklaring:
Multiplicere ud af # Ab # giver os:
# A ^ 2-b ^ 2 = 3ab #
# A ^ 2-3ab-b ^ 2 = 0 #
# A = (3b + -sqrt (9b ^ 2 + 4b ^ 2)) / 2 #
# A = (3b + -sqrt (13b ^ 2)) / 2 #
# A = (3b + -bsqrt (13)) / 2 #
# A = (b (3 + -sqrt13)) / 2 #
# A ^ 3 = ((b (3 + sqrt13)) / 2) ^ 3eller ((b (3-sqrt13)) / 2) ^ 3 #
# A ^ 3 = (b ^ 3 (144 + 40sqrt13)) / 8or (b ^ 3 (144-40sqrt13)) / 8 # (fra binomial ekspansion)
# A ^ 3 = b ^ 3 (18 + 5sqrt13) orb ^ 3 (18-5sqrt13) #
# (B ^ 3 (18 + 5sqrt13)) / b ^ 3 + b ^ 3 / (b ^ 3 (18 + 5sqrt13)) = 18 + 5sqrt13 + 1 / (18 + 5sqrt13) = 10sqrt13 #
# (B ^ 3 (18-5sqrt13)) / b ^ 3 + b ^ 3 / (b ^ 3 (18-5sqrt13)) = 18-5sqrt13 + 1 / (18-5sqrt13) = - 10sqrt13 #
# (B ^ 3 (18 + 5sqrt13)) / b ^ 3 + b ^ 3 / (b ^ 3 (18-5sqrt13)) = 18-5sqrt13 + 1 / (18-5sqrt13) = 0 #
# (B ^ 3 (18-5sqrt13)) / b ^ 3 + b ^ 3 / (b ^ 3 (18 + 5sqrt13)) = 18-5sqrt13 + 1 / (18-5sqrt13) = 0 #
# A ^ 3 / b ^ 3 + b ^ 3 / a ^ 3 = 0or10sqrt13 #
Svar:
# a ^ 3 / b ^ 3 + b ^ 3 / a ^ 3 = + -10sqrt (13) #
Forklaring:
Givet:
# a / b-b / a = 3 #
Squaring begge sider får vi:
# a ^ 2 / b ^ 2-2 + b ^ 2 / a ^ 2 = 9 #
Transponering og tilføjelse #4# til begge sider finder vi:
# 13 = a ^ 2 / b ^ 2 + 2 + b ^ 2 / a ^ 2 = (a / b + b / a) ^ 2 #
Noter det:
# (a / b + b / a) ^ 3 = a ^ 3 / b ^ 3 + 3a / b + 3b / a + b ^ 3 / a ^ 3 #
Så:
# a ^ 3 / b ^ 3 + b ^ 3 / a ^ 3 = (a / b + b / a) ((a / b + b / a) ^ 2-3)
#color (hvid) (a ^ 3 / b ^ 3 + b ^ 3 / a ^ 3) = (a / b + b / a) (13-3)
#color (hvid) (a ^ 3 / b ^ 3 + b ^ 3 / a ^ 3) = + -10sqrt (13) #
Svar:
# = pm 10 sqrt (13) #
Forklaring:
#"Navn"#
#x = a / b, "" y = b / a #
# => xy = 1, "og" x - y = 3 #
# => x = y + 3 #
# => y ^ 2 + 3 y - 1 = 0 #
# => y = (- 3 pm sqrt (13)) / 2 #
# => x = (3 pm sqrt (13)) / 2 #
# x ^ 3 + y ^ 3 = (x + y) (x ^ 2 - x y + y ^ 2) #
# = pm sqrt (13) (22/4 - 1 + 22/4) #
# = pm 10 sqrt (13) #
