Question

How do you solve `(x-2)^(3/4)=8`?

Answer 1

`(x-2)^(3/4)=8`

There is an easy solution which is:

`((x-2)^(3/4))^(4/3)=8^(4/3)`

`(x-2)^1=(2^3)^(4/3)`

`x-2=2^4=16`

`x=18`,

But we can get other solutions:

`(x-2)^(3/4)=8`

`((x-2)^(3/4))^4=8^4` This step is dangerous since we can be adding false solutions.

`(x-2)^3=8^4=4096`

`x^3-6x^2+12x-8=4096`

`x^3-6x^2+12x-4104=0`

We already know that 18 is one of the zeros:

So we divide the third degree polynom by (x-18):

`(x^3-6x^2+12x-4104)/(x-18)=x^2+12x+228`

`x^2+12x+228` can be solved by the quadratic formula:

`x=(-12+-sqrt(12^2-4*228))/2`

`x=(-12+-sqrt(-768))/2`

`x=(-12+-16sqrt(-3))/2`

`x=-6+-8isqrt(3)`

These are false solutions, since `(x-2)^(3/4)=+-8i`. These false solutions were added in the step where it is written :"This step is dangerous since we can be adding false solutions."

So the only solution is x=18.

Answer 2

`x=18`

Assuming no complex solution is required

Explanation:

Given:`" "(x-2)^(color(magenta)(3/4))=8`

Remember that `(x-2)^(3/4)=root(3)((x-2)^4)`

Take the cube root of both sides giving

`" "(x-2)^(color(magenta)(1/4))= 2`

Raise both sides to the power of 4 giving

`x-2=2^4 = 16`

`x=18`