Question

What is the solution to the equation? Please explain the steps to this problem

`16=(x-2)^(2/3`

Answer 1

`x=66`

Explanation:

First, let's get rid of that nasty exponent. An exponent rule we can use is this:
`a^(b/c)=root(c)(a^b)`

Let's use it to simplify the right side of our equation:

`(x-2)^(2/3) = root(3)((x-2)^2)`

`16=root(3)((x-2)^2)`

Next, we need to remove the radical. Let's cube, or apply a power of 3, to each side. Here's how it will work:

`(root(3)(a))^3 = a^(1/3*3)=a^(3/3)=a^1=a`

We will apply this to our equation:

`(16)^3=(root(3)((x-2)^2))^3`
`(16)^3=(x-2)^2`
`4096=(x-2)^2`

Then, we will square each side. It works in the opposite way of the last step:

`sqrt(a^2)=a^(2*1/2)=a^(2/2)=a^1=a`

`sqrt(4096)=sqrt((x-2)^2)`

`64=x-2`

From here, we just add 2 to each side to get our answer:

`64+2=x-2+2`
`66=x`

Hope this helped you! Have a nice day!!!