To solve the equation \( x^{2/3} = 64 \), we will first eliminate the rational exponent by raising both sides of the equation to the reciprocal of \( \frac{2}{3} \), which is \( \frac{3}{2} \).
\[ \left( x^{2/3} \right)^{3/2} = 64^{3/2} \]
On the left side, the exponents multiply:
\[ x^{(2/3) \cdot (3/2)} = x^1 = x \]
Next, we calculate the right side:
\[ 64^{3/2} = (64^{1/2})^3 \]
Calculating \( 64^{1/2} \):
\[ 64^{1/2} = \sqrt{64} = 8 \]
Now we can raise this result to the power of 3:
\[ 8^3 = 512 \]
Putting it all together, we have:
\[ x = 512 \]
Thus, the solution to the equation \( x^{2/3} = 64 \) is
\[ \boxed{512} \]
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