To solve the equation \( x^{2/3} = 64 \), follow these steps:
- Raise both sides to the reciprocal of the exponent: The reciprocal of \( \frac{2}{3} \) is \( \frac{3}{2} \).
\[ (x^{2/3})^{\frac{3}{2}} = 64^{\frac{3}{2}} \]
- Simplify the left side:
\[ x^{(2/3) \cdot (3/2)} = x^1 = x \]
- Calculate the right side:
First, calculate \( 64^{1/2} \):
\[ 64^{1/2} = \sqrt{64} = 8 \]
Now raise that result to the power of 3:
\[ 64^{3/2} = 8^3 = 512 \]
- Write the final equation:
\[ x = 512 \]
Thus, the solution to the equation \( x^{2/3} = 64 \) is
\[ \boxed{512} \]