If #a/b - b/a =3# Then what is the value of #a^3/b^3 + b^3/a^3#?

3 Answers
May 25, 2018

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

Explanation:

Multiplying out by #ab# gives us:
#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)^3or((b(3-sqrt13))/2)^3#

#a^3=(b^3(144+40sqrt13))/8or(b^3(144-40sqrt13))/8# (from binomial expansion)

#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#

May 25, 2018

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

Explanation:

Given:

#a/b-b/a = 3#

Squaring both sides, we get:

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

Transposing and adding #4# to both sides, we find:

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

Note that:

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

So:

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

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

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

May 25, 2018

#= pm 10 sqrt(13)#

Explanation:

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

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