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

3 Answers
Aug 11, 2018

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

Aug 11, 2018

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

Aug 11, 2018

(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