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

Question

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

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Answer 1 · 2018-06-26T01:22:40

#x=66#

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!!!