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To solve the equation \( X^{\frac{2}{3}} = 64 \), we can start by raising both sides of the equation to the reciprocal of the exponent, which is \( \frac{3}{2} \).
\[
X = 64^{\frac{3}{2}}
\]
Now we can simplify \( 64^{\frac{3}{2}} \):
1. First, find the square root of 64:
\[
\sqrt{64} = 8
\]
2. Then raise it to the power of 3:
\[
8^3 = 512
\]
Thus, we have:
\[
X = 512
\]
So the solution to the equation \( X^{\frac{2}{3}} = 64 \) is:
\(\boxed{512}\)