Answer:
[tex]x=513 \Rightarrow S=\{513\}[/tex]
Step-by-step explanation:
1) Rewriting for the sake of clarity:
[tex](x-1)^{\frac{2}{3}}=64[/tex]
2) To solve it, we need to turn the rational exponent equal to 1 via exponent rule of product. Multiplying 2/3 by its reciprocal 3/2 we get 1, since this is an equality we need to do it on the other side.
[tex](x-1)^{\frac{2}{3}}=64\\\\((x-1)^{\frac{2}{3}}]^{\frac{3}{2}}=(64)^{\frac{3}{2}}\\(x-1)=\sqrt{64^{3}}\\x-1+1=512+1\\x=513[/tex]
