Question

What is the value of x in the equation (a/b)^(2x-3) = (b^3/a^3)^(x+4)?

Answer 1

`x=-9/5`

Explanation:

Provided information is

`(a/b)^(2x-3) = (b^3/a^3)^(x+4)`

so we can rearrange it as,

`(a/b)^(2x-3)=(b/a)^(3(x+4))=(b/a) ^(3x+12)=(a/b)^(-3x-12)`

{ as, `1/b^(-x) =b^0/b^-x=b^(0-(-x))=b^x` similarly, `1/a^x` can be written as `a^0/a^x=a^(0-x)=a^-x`}

Now,comparing,both sides,

`2x-3=-3x-12`

or, `5x=-9`

or, `x=-9/5`

Answer 2

`x = -9/5`

Explanation:

`(a/b)^(2x-3) = (b^3/a^3)^(x+4)`

Recall;

`color(indigo)((x^a/y^a) = (x/y)^a)`

Hence;

`(a/b)^(2x-3) = (b/a)^(3(x+4)`

Simplifying;

`(a/b)^(2x-3) = (b/a)^(3x+12)`

Also recall;

`color(indigo)(y/x = 1/(x/y) or (x/y)^-1)`

Hence;

`(a/b)^(2x-3) = 1/(a/b)^(3x+12)`

Simplifying;

`(a/b)^(2x-3) = (a/b)^(-1(3x+12)`

`(a/b)^(2x-3) = (a/b)^(-3x-12)`

`cancel((a/b))^(2x-3) = cancel((a/b))^(-3x-12)`

`2x - 3 = -3x - 12`

Collecting like terms;

`2x + 3x = -12 + 3`

`5x = -9`

`x = -9/5`

Answer 3

`(a/b)^(2x-3) = (b^3/a^3)^(x+4)`

`=>a^(2x-3)xxa^(3x+12) = b^(3x+12)xxb^(2x-3)`

`=>a^(5x+9) = b^(5x+9)`

`=>a^(5x+9) / b^(5x+9)=1`

`=>(a/b)^(5x+9) =(a/b)^0`

`=>5x+9=0`

`=>x=-9/5`