Question

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

Answer 1

`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`

Answer 2

`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)`

Answer 3

`= 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)`