Als a / b - b / a = 3 Wat is dan de waarde van een ^ 3 / b ^ 3 + b ^ 3 / a ^ 3?

Antwoord:

# A ^ 3 / b ^ 3 + b ^ 3 / a ^ 3 = 0or10sqrt13 #

Uitleg:

Vermenigvuldigen met # Ab # geeft ons:

# 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) ^ 3of ((bis (3-sqrt13)) / 2) ^ 3 #

# A ^ 3 = (b ^ 3 (144 + 40sqrt13)) / 8of (b ^ 3 (144-40sqrt13)) / 8 # (van binomiale uitbreiding)

# A ^ 3 = b ^ 3 (18 + 5sqrt13) orb ^ 3 (18-5sqrt13) #

# (B ^ 3 (18 + 5sqrt13)) / b ^ 3 ^ 3 + b / (b ^ 3 (18 + 5sqrt13)) = 18 + 5sqrt13 + 1 / (18 + 5sqrt13) = 10sqrt13 #

# (B ^ 3 (18-5sqrt13)) / b ^ 3 ^ 3 + b / (b ^ 3 (18-5sqrt13)) = 18-5sqrt13 + 1 / (18-5sqrt13) = - 10sqrt13 #

# (B ^ 3 (18 + 5sqrt13)) / b ^ 3 ^ 3 + b / (b ^ 3 (18-5sqrt13)) = 18-5sqrt13 + 1 / (18-5sqrt13) = 0 #

# (B ^ 3 (18-5sqrt13)) / b ^ 3 ^ 3 + b / (b ^ 3 (18 + 5sqrt13)) = 18-5sqrt13 + 1 / (18-5sqrt13) = 0 #

# A ^ 3 / b ^ 3 + b ^ 3 / a ^ 3 = 0or10sqrt13 #

Antwoord:

# a ^ 3 / b ^ 3 + b ^ 3 / a ^ 3 = + -10sqrt (13) #

Uitleg:

Gegeven:

# a / b-b / a = 3 #

Aan beide kanten kwadraten, krijgen we:

# a ^ 2 / b ^ 2-2 + b ^ 2 / a ^ 2 = 9 #

Omzetten en toevoegen #4# aan beide kanten vinden we:

# 13 = a ^ 2 / b ^ 2 + 2 + b ^ 2 / a ^ 2 = (a / b + b / a) ^ 2 #

Let daar op:

# (a / b + b / a) ^ 3 = a ^ 3 / b ^ 3 + 3a / b + 3b / a + b ^ 3 / a ^ 3 #

Zo:

# a ^ 3 / b ^ 3 + b ^ 3 / a ^ 3 = (a / b + b / a) ((a / b + b / a) ^ 2-3) #

#color (wit) (a ^ 3 / b ^ 3 + b ^ 3 / a ^ 3) = (a / b + b / a) (13-3) #

#color (wit) (a ^ 3 / b ^ 3 + b ^ 3 / a ^ 3) = + -10sqrt (13) #

Antwoord:

# = pm 10 sqrt (13) #

Uitleg:

#"Naam"#

#x = a / b, "" y = b / a #

# => xy = 1, "en" 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) #