To solve the equation \( x^{2/3} = 64 \), we can first eliminate the rational exponent by raising both sides of the equation to the reciprocal of \(\frac{2}{3}\), which is \(\frac{3}{2}\).
\[ \left(x^{2/3}\right)^{\frac{3}{2}} = 64^{\frac{3}{2}} \]
This simplifies to:
\[ x^{(2/3) \cdot (3/2)} = 64^{\frac{3}{2}} \]
Since \( (2/3) \cdot (3/2) = 1 \), we have:
\[ x = 64^{\frac{3}{2}} \]
Now, let's calculate \( 64^{\frac{3}{2}} \):
-
First, calculate the square root of 64: \[ \sqrt{64} = 8 \]
-
Then, cube the result: \[ 8^3 = 512 \]
Now, we can conclude that:
\[ x = 512 \]
Thus, the solution to the equation \( x^{2/3} = 64 \) is \( \boxed{512} \).