x = -22 and 32 are your solutions.
here's why:
(x-5)^(2/3)= 9 is the original equation.
(x-5)^(2/3) is the same as ((x-5)^2)^(1/3)
your equation becomes:
((x-5)^2)^(1/3) = 9
you want to get rid of the cube root, so you cube both sides of the equation.
you get:
(((x-5)^2)^(1/3))^3 = 9^3
simplify this and you get:
(x-5)^2 = 9^3 which becomes (x-5)^2 = 729
simplify (x-5)^2 by expanding it and you get:
x^2 - 10x + 25 = 729
subtract 25 from both sides of the equation to get:
x^2 - 10x - 704 = 0
factor this to get:
(x+22) * (x-32) = 0
solve for x to get the possible solutions of:
x = -22 or x = 32
confirm your solutions are good by replacing x in the oridinal equations to see if the equations hold true.
this step is necessary since there are problems where the possible solutions are not feasible.
the original equation is:
(x-5)^(2/3)= 9
replace x with -22 and you get:
(-22-5)^(2/3) = 9 which becomes (-27)^(2/3) = 9
(-27)^(2/3) is equivalent to either:
(-27)^2 = 729 and the cube root of 729 is eqjual to 3.
alternatively:
cube root of (-27) = (-3) and (-3)^2 = 9
replace x with 32 and you get:
((32-5)^(2/3) = 9 which becomes ((27)^(2/3) = 9
27 squared = 729 and the cube root of 729 is equal to 9
alternatively:
cube root of 27 = 3 and 3 squared = 9.
looks like both solutions are good.
i graphed the equation of y = (x-5)^(2/3) and the equation of y = 9 to confirm graphically that this is true.
in the graph, you can see the intersection points of the 2 equations are at x = -22 and x = 32.
that confirms the solution is correct graphically.
the graph is shown below:
